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+1
+Scalable Optimal Design of Incremental Volt/VAR
+Control using Deep Neural Networks
+Sarthak Gupta, Graduate Student Member, IEEE, Ali Mehrizi-Sani, Senior Member, IEEE,
+Spyros Chatzivasileiadis, Senior Member, IEEE, and Vassilis Kekatos, Senior Member, IEEE
+Abstract—Volt/VAR control rules facilitate the autonomous
+operation of distributed energy resources (DER) to regulate
+voltage in power distribution grids. According to non-incremental
+control rules, such as the one mandated by the IEEE Standard
+1547, the reactive power setpoint of each DER is computed as a
+piecewise-linear curve of the local voltage. However, the slopes
+of such curves are upper-bounded to ensure stability. On the
+other hand, incremental rules add a memory term into the
+setpoint update, rendering them universally stable. They can
+thus attain enhanced steady-state voltage profiles. Optimal rule
+design (ORD) for incremental rules can be formulated as a bilevel
+program. We put forth a scalable solution by reformulating ORD
+as training a deep neural network (DNN). This DNN emulates the
+Volt/VAR dynamics for incremental rules derived as iterations of
+proximal gradient descent (PGD). The rule parameters appear as
+DNN weights. To reduce the DNN depth, we leverage Nesterov’s
+accelerated PGD iterations. Analytical findings and numerical
+tests corroborate that the proposed ORD solution can be neatly
+adapted to single/multi-phase feeders.
+Index Terms—IEEE Standard 1547.8, incremental control
+rules, multiphase feeders, proximal gradients, gradient backprop-
+agation, deep neural networks.
+I. INTRODUCTION
+Local Volt/VAR (Volt-Ampere Reactive) control facilitates
+voltage regulation on distribution grids by providing reactive
+power compensation from DERs equipped with smart invert-
+ers. Different from centralized control schemes which incur
+large computational and communication burden, local rules
+decide DER setpoints based on local measurements. Volt/VAR
+control rules can be categorized into non-incremental and
+incremental ones. The former compute DER reactive power
+setpoints based on local voltage readings. The IEEE Stan-
+dard 1547.8 prescribes such non-incremental control rules
+as piecewise-linear functions of voltage [1]. On the other
+hand, incremental Volt/VAR rules compute the change in VAR
+setpoints as a function of voltage [2]–[6].
+The existing literature on designing Volt/VAR control rules
+can be classified into stability- and optimality-centric works.
+Stability-centric works study the effect of Volt/VAR rules as a
+closed-loop dynamical system, which may be rendered unsta-
+ble under steep slopes of non-incremental rules [7], [8]. In fact,
+to ensure stability, non-incremental rules may have to compro-
+mise on the quality of their steady-state voltage profile [5],
+[8]. Incremental rules however do not experience stability
+limitations and can thus achieve improved voltage profiles
+compared to their non-incremental counterparts. Nonetheless,
+such improvements may come at the expense of longer settling
+times of the associated Volt/VAR dynamics [8].
+Optimality-centric works focus on designing stable control
+rules to minimize a voltage regulation objective. To this end,
+optimization-based strategies have been employed to design
+affine non-incremental rules using heuristics [9]–[11]. Two of
+our recent works in [12] and [13] have addressed the problem
+of optimally designing the slope, deadband, saturation, and
+reference voltage. Reference [12] performs ORD via a bilevel
+optimization applicable to single-phase feeders. Reference [13]
+proposes DNN-based digital twins that emulate Volt/VAR
+dynamics, and reformulates ORD as a DNN training task for
+single-/multi-phase feeders.
+This letter deals with optimally selecting the shape of
+incremental Volt/VAR control rules, with contributions on
+three fronts: c1) Although this optimal rule design (ORD)
+task can be posed as a mixed-integer nonlinear optimization
+program, it does not scale well with the numbers of DERs,
+nodes, and grid loading scenarios. To address this challenge,
+the genuine idea here is to reformulate ORD as a deep-
+learning task and judiciously adapt the fast software modules
+widely available for training deep neural networks (DNNs).
+We have put forth a similar approach for designing non-
+incremental control rules in [13]. However, migrating from
+non-incremental to incremental rules is non-trivial due to the
+different curve shapes, stability, and settling time properties.
+c2) To further expedite ORD for incremental rules, we suggest
+implementing accelerated Nesterov-type variants of the rules
+to yield a shallower DNN emulator. c3) We also establish the
+convergence of incremental rules on multiphase feeders.
+Recently, reference [14] deals with the optimal design of
+incremental rules. It uses DNNs with a single hidden layer
+to model piecewise-linear functions and formulates ORD as
+a reinforcement learning task. While [14] also utilizes DNNs
+to design incremental rules, we delineate from it in several
+ways. Reference [14] focuses on voltage control during tran-
+sient dynamics, whereas this work aims at ORD to drive
+steady-state voltages closer to unity and over different grid
+loading scenarios. Reference [14] utilizes a DNN to model
+the piecewise-linear mapping of the rule. In contrast, this
+work develops a DNN-based digital twin that emulates end-
+to-end Volt/VAR dynamics. Lastly, we provide stability and
+convergence analysis for single- and multiphase feeders alike,
+whereas [14] applies only to single-phase feeders.
+The rest of this letter is organized as follows. Section II
+models the feeder and discusses non-incremental and incre-
+mental Volt/VAR control rules. Section III formulates DNN-
+based digital twins for Volt/VAR dynamics of incremental
+rules, and their accelerated version. It also presents ORD
+arXiv:2301.01440v1  [math.OC]  4 Jan 2023
+
+2
+Fig. 1. Non-incremental Volt/VAR control rule provisioned by the IEEE Std.
+1547 for the interconnection of DERs [1].
+for single-phase feeders as a deep learning task. Section IV
+extends the ORD process to multiphase feeders. The incre-
+mental rules are then benchmarked against non-incremental
+rules from [13] using tests on real-world data, in Section V.
+The letter is concluded in Section VI.
+II. VOLT/VAR CONTROL RULES
+Consider a radial feeder serving N buses equipped with
+DERs, indexed by n. Let (qℓ, q) collect reactive loads and
+generations at all nodes. Vectors (p, v) collect the net active
+power injections and voltage magnitudes at all nodes. The
+impact of q on v can be approximately captured using the
+linearized grid model [13]
+v ≃ Xq + ˜v
+(1)
+where ˜v := Rp − Xqℓ + v01 models the underlying grid
+conditions, and v0 is the substation voltage. Vector ˜v rep-
+resents the impact of non-controlled quantities (p, qℓ) on
+voltages. Matrices (R, X) depend on the feeder topology. For
+single-phase feeders, they are symmetric positive definite with
+positive entries [15]. For multiphase feeders, they are non-
+symmetric and have positive and negative entries [5], [13].
+Vector q in (1) carries the reactive injections by DERs
+we would like to control. Per the non-incremental rules of
+the IEEE Std. 1547 [1], DER setpoints are decided based
+on the Volt/VAR curve of Fig. 1, which is parameterized by
+(¯v, δ, σ, ¯q). The standard further constrains these parameters
+within a polytopic feasible set [1], [12]. The negative slope of
+the linear segment of the curve in Fig. 1 can be expressed as
+α :=
+¯q
+σ − δ .
+The interaction of Volt/VAR rules with the feeder gives
+rise to nonlinear dynamics. These dynamics are stable if
+∥ dg(α)X∥2 < 1, where dg(α) is a diagonal matrix carry-
+ing the rule slopes over all buses on its diagonal [7]. The
+equilibrium setpoints for DERs cannot be expressed in closed
+form. However, they coincide with the minimizer of the convex
+optimization problem [7]
+min
+−¯q≤q≤¯q
+1
+2q⊤Xq+q⊤(˜v−¯v)+ 1
+2q⊤ dg−1(α)q+δ⊤|q| (2)
+where |q| applies the absolute value on q entrywise. Prob-
+lem (2) depends on rule parameters (¯v, δ, α, ¯q) across all buses,
+collected in the 4N-long vector z := (¯v, δ, α, ¯q). We denote
+by qz(˜v) the equilibrium setpoints, and by
+vz(˜v) = Xqz(˜v) + ˜v
+(3)
+the related equilibrium voltages reached by Volt/VAR rules
+parameterized by z under grid conditions ˜v.
+Optimal rule design (ORD) can be stated as the task of
+selecting z to bring equilibrium voltages vz(˜v) close to unity.
+To cater to diverse conditions, the utility may sample loading
+scenarios {˜vs}S
+s=1 for the next hour, and find z as
+z∗ ∈ arg min
+z
+F(z) := 1
+S
+S
+�
+s=1
+∥vz(˜vs) − 1∥2
+2
+(ORD)
+subject to (3) and z ∈ Z.
+Once found, the customized rules z∗ are sent to DERs to
+operate autonomously over the next hour. Note that vz(˜vs)
+depends on z because the equilibrium setpoints qz(vs) in (3)
+are the minimizers of problem (2), which is parameterized by
+z. When solving (ORD) for non-incremental rules, the feasible
+set Z consists of the polytopic constraints imposed on z by
+the IEEE Std. 1547 as well as additional constraints on α
+to ensure ∥ dg(α)X∥2 < 1; see [12]. Therefore, the feasible
+set Z can be quite confined. This can lead to less desirable
+voltage profiles; that is, higher objective values F(z∗).
+The aforesaid issue can be addressed by replacing the non-
+incremental Volt/VAR rules of IEEE Std. 1547 by incremental
+ones as suggested in [2]–[6]. Incremental rules express the
+change rather than the actual value in setpoints as a function
+of voltage. One option for incremental rules is to implement
+a proximal gradient descent (PGD) algorithm solving (2)
+as proposed in [5]. In this case, the control rule coincides
+with the PGD iterations, which are implemented by DERs
+in a decentralized fashion. Using incremental rules, set Z is
+enlarged as now we only need to ensure
+z ≥ 0
+0.95 · 1 ≤ ¯v ≤ 1.05 · 1
+and that ¯q are within the reactive power ratings of the DERs.
+The PGD algorithm is an extension of gradient descent to
+handle constraints and non-differentiable costs [5]. At iteration
+t, PGD proceeds with two steps: s1) It first computes the gradi-
+ent of the first two terms of F(z), that is Xqt+˜v−¯v = vt−¯v.
+Here qt is the latest estimate of the minimizer of (2); s2) PGD
+then updates qt+1 as the minimizer of
+min
+−¯q≤q≤¯q
+1
+2q⊤ dg−1(α)q + δ⊤|q| + 1
+2µ∥q − (vt − ¯v)∥2
+2 (4)
+for a step size µ > 0. The last problem involves the last two
+terms in the cost of (2) regularized by the Euclidean distance
+of q to the gradient (vt − ¯v) computed in step s1).
+Converting PGD to control rules, step s1) is performed by
+the physics of the feeder when injecting qt and measuring
+the local voltage deviations. Step s2) is run by each DER
+independently as (4) is separable across buses. Using the
+subdifferential, solving (4) provides the update [5]
+yt
+n = ˜αn ·
+�
+qt
+n − µ(vt
+n − ¯vn)
+�
+(5a)
+
+fq
+UA3
+qt+1
+n
+= gn
+�
+yt
+n
+�
+(5b)
+where gn(yn) is the proximal operator
+gn(yn) :=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
++¯qn
+, yn > qn + µ˜δn
+yn − µ˜δn
+, µ˜δn < yn ≤ qn + µ˜δn
+0
+, − µ˜δn ≤ yn ≤ µ˜δn
+yn + µ˜δn
+, − qn − µ˜δn ≤ yn < −µ˜δn
+−qn
+, yn < −qn − µ˜δn.
+(6)
+and the new parameters (˜αn, ˜δn) are defined as
+˜αn :=
+1
+1 + µ/αn
+and
+˜δn :=
+δn
+1 + µ/αn
+.
+The proximal operator is plotted in the top panel of Figure 2.
+Note that in (5), rule parameters are transformed from repre-
+sentation z = (¯v, δ, α, ¯q) to representation ˜z := (¯v, ˜δ, ˜α, ¯q).
+This is without loss of generality as the transformation is a
+bijection, and so one can work exclusively with ˜z. The feasible
+set ˜Z for ˜z is similar to Z with the addition that ˜α ≤ 1. As
+with non-incremental rules, the rules in (6) are driven by local
+data, but now qt+1
+n
+depends on (vt
+n, qt
+n), and not vt
+n alone. Both
+types of rules solve (2). Hence, they both converge to the same
+equilibrium. The advantage of incremental rules is that they are
+stable for all α as long as µ < 2/λmax(X); see [5]. It is worth
+stressing that z does not have the same physical interpretation
+as in non-incremental rules (slopes, deadband, or saturation),
+though z parameterizes (2) for both rules.
+Accelerated incremental rules. Although PGD rules en-
+large Z, their settling times can be long. They reach an
+ε-optimal cost of (2) within − 2 log ε
+log 2 κ (X) iterations. Here
+κ(X) := λmax(X)/λmin(X) is the condition number of
+X. References [5], [16] put forth accelerated incremental
+rules based on accelerated PGD (APGD). These rules need
+− 2 log ε
+log 2
+�
+κ (X) iterations to attain an ε-optimal cost, and take
+the form
+˜yt
+n := (1 + βt) yt
+n − βtyt−1
+n
+(7a)
+qt+1
+n
+:= gn
+�
+˜yt
+n
+�
+(7b)
+where βt := t−1
+t+2, while yt
+n and gn(yn) are as defined in (5a)
+and (6). Updates (7) remain local, but introduce additional
+memory as qt+1
+n
+depends on (vt
+n, qt
+n) and (vt−1
+n
+, qt−1
+n
+).
+III. DEEP LEARNING FOR OPTIMAL RULE DESIGN (ORD)
+IN SINGLE-PHASE FEEDERS
+Solving (ORD) is challenging as it is a nonconvex bilevel
+program. Although it can be modeled as a mixed-integer
+nonlinear program, such an approach does not scale well with
+the number of DERs and/or scenarios for non-incremental
+rules [13]. Seeking a more scalable solution, we reformulate
+(ORD) as a deep learning task. The key idea is to design
+a DNN that emulates Volt/VAR dynamics under the control
+rule of (5). To this end, note that gn(yn) is a piecewise-
+linear function with four breakpoints [5]. Interestingly, this
+operator can be expressed as the superposition of four rectified
+linear units (ReLUs) as illustrated in Fig. 2, where ReLUs are
+denoted by ρ(·). The intercepts of the ReLUs depend linearly
+on (˜δn, ¯qn).
+Fig. 2. Proximal operator g(y) expressed as a sum of four shifted rectified
+linear units (ReLUs).
+Fig. 3.
+A DNN emulating the accelerated incremental rules of (7). Plain
+incremental rules can be modeled by dropping the second layer (setting βt =
+0) and ignoring output yt
+n.
+Building on this, one APGD iteration for DER n can be
+implemented by the 4-layer DNN in Fig. 3, whose weights
+depend affinely on (¯vn, ˜δn, ˜αn, ¯qn). This DNN takes (qt
+n, vt
+n)
+as its input, and computes (qt+1
+n
+, yt
+n) at its output. It is
+termed ICn and will be used as a building block to emulate
+Volt/VAR dynamics. This is accomplished by the recursive
+neural network (RNN) shown in Fig. 4. Here blocks ICn are
+arranged vertically to model the parallel operation of DERs.
+Their outputs qt+1 are multiplied by X, and the new voltage
+is computed as vt+1 = Xqt+1 + ˜v. This is repeated T times.
+Thanks to the RNN structure, there is weight sharing, so the
+number of DNN weights is 4N rather than 4NT.
+The RNN takes a grid loading vector ˜vs as its input, the rule
+parameters ˜z as weights, and computes the voltages vT
+˜z (˜vs)
+at time T at its output. For the output vT
+˜z (˜vs) to approximate
+well equilibrium voltages, the depth T can be chosen by the
+convergence rate of PGD as follows.
+Proposition 1. For the DNN of Fig. 4 to ensure ∥Φ (˜v; z) −
+v∗(z)∥2 ≤ ϵ1 ∀ ˜v, its depth T should satisfy
+T ≥
+�κ − 1
+2
+�
+log
+�2∥X∥2∥ˆq∥2
+ϵ1
+�
+.
+(8)
+
++qH
+b+
+qp
+-uon
+一
+qn
+t
+p
+t
+p
+-uon
+p4
+Fig. 4.
+Recurrent neural network (RNN) implementation for accelerated
+incremental Volt/VAR control rules.
+Proof: From the control rule of (5b), it follows that
+∥qt − q∗∥2 = ∥g
+�
+yt�
+− g (y∗) ∥2
+≤ ∥yt − y∗∥2
+= ∥ dg(˜α)(I − µX)
+�
+qt−1 − q∗�
+∥2
+≤ ∥ dg(˜α)∥2 · ∥I − µX∥2 · ∥qt−1 − q∗∥2
+≤ ∥I − µX∥2 · ∥qt−1 − q∗∥2.
+(9)
+The first inequality stems from the non-expansive property of
+the proximal operator g. The next equality follows from (5a).
+The second inequality from the sub-multiplicative property of
+the spectral norm. The last inequality follows by the definition
+of spectral norm and because ˜αn ≤ 1 for all n.
+If ∥I − µX∥2 < 1, inequality (9) implies that the dynamics
+in (5) are a non-expansive mapping, and thus, are stable and
+converge to q∗. Condition ∥I − µX∥2 < 1 holds when µ <
+2/λmax(X). The norm ∥I − µX∥2 achieves its minimum of
+�
+1 −
+2
+κ+1
+�
+when
+µ0 :=
+2
+λmax(X) + λmin(X).
+Plugging µ0 into (9) and unfolding the dynamics over t
+provides
+∥qt − q∗∥2 ≤
+�
+1 −
+2
+κ+1
+�t
+∥q0 − q∗∥2
+≤ 2
+�
+1 −
+2
+κ+1
+�t
+∥ˆq∥2.
+For
+the
+voltage
+approximation
+error
+∥vT − v∗∥2
+=
+∥X
+�
+qT − q∗�
+∥2 at time T to be smaller than ϵ1, we need
+∥vT − v∗∥2 ≤ 2∥X∥2 · ∥ˆq∥2 ·
+�
+1 −
+2
+κ + 1
+�T
+≤ ϵ1.
+This can be achieved by selecting T such that
+T ≥
+log
+�
+2∥X∥2∥ˆq∥2
+ϵ1
+�
+log
+�
+1 +
+2
+κ−1
+�
+≥
+�κ − 1
+2
+�
+log 2∥X∥2∥ˆq∥2
+ϵ1
+.
+where the last inequality follows from log(1 + x) ≤ x.
+Plugging the values ∥X∥2 = 0.463 and κ = 848 for the
+IEEE 37-bus feeder, ∥ˆq∥2 = 0.1, and ϵ1 = 10−5 in (8), yields
+T ≥ 2, 892 layers, which is relatively large. A key contributor
+to this large T is the κ term in (8). This promulgates the
+adoption of accelerated rules (7), which are known to have
+O(√κ) dependence. Interestingly, during implementation, one
+does not need to fix T to the above worst-case bounds.
+Leveraging dynamic computation graphs offered by Python
+libraries such as Pytorch, one may determine T ‘on the
+fly’ depending on the convergence of vt between pairs of
+successive layers.
+Since the RNN emulates Volt/VAR dynamics, it can surro-
+gate vz(˜vs) in (ORD). Then (ORD) can be posed as training
+a DNN over its weights ˜z ∈
+˜Z or z ∈ Z. Grid loading
+scenarios {˜vs}S
+s=1 are treated as features and equilibrium
+voltages vz(˜vs) as predictions that should be brought close to
+the target value of 1 for scenarios s. The DNN can be trained
+using stochastic projected gradient descent (SPGD) as [13]
+˜zi+1 =
+�
+˜zi − λ
+2B ∇˜zi
+� �
+s∈Bi
+∥Φ(˜vs; ˜z) − 1∥2
+2
+��
+˜
+Z
+(10)
+where λ > 0 is the learning rate; set Bi is a batch of
+B scenarios; and [·] ˜
+Z is the projection onto
+˜Z. Since
+˜Z
+consists of simple box constraints, projection essentially means
+clipping the values to the box. Lastly ∇˜zi(·) represents the
+gradient with respect to ˜z evaluated at ˜z = ˜zi, and is
+calculated efficiently thanks to gradient back-propagation.
+Although our DNN-based ORD assumed PGD-based rule, it
+may be applicable to other incremental rules too.
+A discussion about control rules and their DNN-based
+emulators is due. Recall that all three types of Volt/VAR
+control rules (non-incremental, incremental, and accelerated
+incremental) reach the same equilibrium voltages, if stable.
+The emulators aim at computing these equilibrium voltages.
+A natural question is whether the DNN emulator could im-
+plement a rule of a type different from the rule actually
+implemented on the feeder. This may be desirable to leverage
+the advantages of two types. Some caution is needed here. If
+the feeder implements non-incremental rules, but incremental
+rules converge faster to equilibrium voltages, it makes sense
+for the emulator to implement incremental rules. Of course,
+in this case, stability constraints on the non-incremental rules
+have to be enforced during DNN training. The reverse is not
+recommended: If the emulator implements non-incremental
+rules, its parameters z should be constrained to be stable and
+that would be a restriction of the actual ORD problem. Finally,
+given the convergence advantage of accelerated incremental
+rules, they are always preferable over plain incremental rules
+for the DNN implementation. This showcases the utility of
+accelerated control rules even if they are not actually imple-
+mented on the feeder.
+IV. DEEP LEARNING FOR OPTIMAL RULE DESIGN (ORD)
+IN MULTIPHASE FEEDERS
+In multiphase feeders, matrix X is non-symmetric and has
+both positive and negative entries. Therefore, the rule analysis
+and design of Section III has to be revisited. For example,
+equilibrium setpoints cannot be found as the minimizers of
+
+IC1
+七
+q
+t+1
+q
+IC2
+t
+-
+t
+t-1
+IC N5
+an optimization problem as with (2). Moreover, increasing q
+does not mean that all voltages increase.
+In multiphase feeders, the non-incremental rules of IEEE
+Std. 1547 remain stable as long as ∥ dg(α)X∥2 < 1. This is
+the same condition as in the single-phase setup. How about
+the stability and equilibrium of incremental rules in multiphase
+feeders? Recall that for single-phase rules, incremental rules
+were obtained as the PGD iterations solving (2). Lacking an
+equivalent inner optimization for multiphase feeders precludes
+a similar approach here. Despite the incremental rules of (5)
+do not correspond to PGD iterates anymore, they can still be
+shown to be stable for multiphase feeders.
+Proposition 2. Let UΛU⊤ be the eigen-decomposition of
+matrix XX⊤. The incremental rules of (5) are stable for
+multiphase feeders if their step size is selected as µ <
+λmin
+�
+Λ−1/2U⊤ �
+X + X⊤�
+UΛ−1/2�
+.
+The claim follows readily by adopting the proof of Proposi-
+tion 1: If µ is selected as above, then ∥I − µX∥2 < 1 follows
+from [5, Prop. 6]. Similar to the single-phase case, incremental
+rules in multiphase feeders allow us to enlarge the feasible
+set Z of rule parameters z. It is worth stressing that different
+from the single-phase setting, incremental and non-incremental
+rules do not converge to the same equilibrium on multiphase
+feeders.
+The ORD task for multiphase feeders can also be formulated
+as a deep-learning task, with some modifications. Firstly,
+matrices R and X need to be altered. Secondly, the DNNs
+for multiphase feeders have 12N trainable parameters, since
+each layer consists of 3N building modules corresponding
+to bus/phase (node) combinations. Lastly, the step size has
+to be selected per Proposition 2. Adopting the proof of
+Proposition 1, we next find the minimum DNN depth in
+multiphase feeders.
+Proposition 3. Let the DNN of Fig. 4 implement the incre-
+mental rules of (5) on multiphase feeders with µ selected per
+Proposition 2. The DNN depth T ensuring voltage approxi-
+mation error ∥Φ (˜v; z) − v∗(z)∥2 ≤ ϵ1 is
+T ≥
+log
+ϵ1
+2∥X∥2∥ˆq∥2
+log ∥I − µX∥2
+.
+We next numerically evaluate the proposed DNN-based
+ORD approach in single- and multiphase feeders, and contrast
+the performance of incremental control rules with that of non-
+incremental rules.
+V. NUMERICAL TESTS
+We benchmark the performance of DNN-based incremental
+rules against non-incremental rules from [13] on single- and
+multiphase feeders. Real-world data were sourced from the
+Smart* project on April 2, 2011 [17], as explained in [13].
+The DNNs were implemented and trained using Pytorch.
+We first compare (non)-incremental rules, both designed
+via DNN training for the single-phase IEEE 37-bus feeder
+of Figure 5. Homes with IDs 20–369 were averaged 10
+at a time and successively added as active loads to buses
+2–26 as shown in Fig. 6. Active generation from solar
+Fig. 5.
+The IEEE 37-bus feeder converted to single-phase. Node num-
+bering follows the format node number {panel ID}. DERs at buses
+{6, 9, 11, 12, 15, 16, 20, 22, 24, 25} provide reactive power control; the rest
+operate at unit power factor.
+TABLE I
+INCREMENTAL VS. NON-INCREMENTAL VOLT/VAR CONTROL RULES ON
+THE SINGLE-PHASE IEEE 37-BUS FEEDER
+Time
+q = 0
+Non-incremental
+Incremental
+Obj. (p.u.)
+Time (s)
+Obj. (p.u.)
+Time (s)
+Obj. (p.u.)
+1 pm
+3.01 · 10−3
+37.98
+3.68 · 10−4
+39.39
+3.66 · 10−4
+2 pm
+3.13 · 10−3
+42.93
+4.26 · 10−4
+37.91
+4.25 · 10−4
+3 pm
+4.24 · 10−3
+45.02
+8.59 · 10−4
+34.97
+8.50 · 10−4
+4 pm
+2.12 · 10−3
+48.30
+1.47 · 10−4
+38.52
+1.48 · 10−4
+5 pm
+8.53 · 10−4
+47.37
+9.70 · 10−5
+374.01
+6.90 · 10−5
+panels was also added, as per the mapping in Fig. 6.
+Buses {6, 9, 11, 12, 15, 16, 20, 22, 24, 25} were assumed to
+host DERs with Volt/VAR control customized per bus.
+Incremental rules were simulated in their accelerated ren-
+dition. Both sets of rules were trained over S = 80 scenarios
+and 200 epochs with a learning rate of 0.001, using the
+Adam optimizer, and setting µ = 1 for incremental rules. To
+ensure repeatability, the results were repeated across several
+time periods between 1–6 PM, and are compiled in Table V.
+Incremental rules obtained marginally lower objectives than
+non-incremental rules across all periods, with a somewhat
+significant difference for the 5 PM period. This behavior is
+explained because incremental rules allow for a larger set Z.
+DNN-based incremental control rules were also contrasted
+with their non-incremental ones on the multiphase IEEE 13-
+bus feeder, using the testing setup from [13]. Active loads were
+sampled 10 at a time from homes with IDs 20-379 and added
+to all three phases for the buses 1-12. Figure 6 also shows the
+solar panel assignments shown in Fig 6 for solar generation.
+Lastly, nine DERs with inverters were added across phases
+and bus indices as shown in Fig 6.
+The learning rates for non-incremental and incremental
+DNNs were set as 0.1 and 0.001, respectively, with the design
+
+Z
+Z
+Z
+Z
+Z
+Z6
+Fig. 6. Multiphase IEEE 13-bus distribution feeder.
+TABLE II
+INCREMENTAL VS. NON-INCREMENTAL VOLT/VAR CONTROL RULES ON
+THE MULTIPHASE IEEE 13-BUS FEEDER
+Time
+q = 0
+Non-incremental
+Incremental
+Obj. (p.u.)
+Time (s)
+Obj. (p.u.)
+Time (s)
+Obj. (p.u.)
+1 pm
+2.51 · 10−3
+64.65
+1.15 · 10−3
+199.24
+4.11 · 10−4
+2 pm
+1.48 · 10−3
+66.60
+6.89 · 10−4
+209.92
+3.03 · 10−4
+3 pm
+6.89 · 10−4
+74.68
+4.94 · 10−4
+263.37
+2.16 · 10−4
+4 pm
+8.03 · 10−4
+68.32
+5.26 · 10−4
+126.81
+2.47 · 10−4
+5 pm
+5.51 · 10−4
+62.58
+4.11 · 10−4
+129.71
+1.95 · 10−4
+parameters z := (¯v, δ, σ, α) initialized to feasible values
+(0.95, 0.01, 0.3, 1.5). Table V compares the performance of
+the two rule categories over multiple periods for S = 80.
+While incremental rules took longer times to train, they were
+successful in lowering the cost F(z) by more than 50%, thus
+yielding improved voltage profiles across all periods.
+VI. CONCLUSIONS
+We have devised a DNN approach to optimally design
+incremental Volt/VAR control rules for single- and multi-phase
+feeders. The key idea is to construct a DNN that emulates
+end-to-end the associated Volt/VAR dynamics. The DNN takes
+grid conditions as the input, the rule parameters as weights,
+and outputs the associated equilibrium voltages. Leveraging
+the convergence rates of the related optimization algorithms,
+we have provided bounds on the minimum depth of the
+DNN emulator to approximate equilibrium voltages within the
+desired accuracy. We have also established the stability of
+incremental control rules for multiphase feeders. Numerical
+tests have demonstrated that the designed control rules attain
+improved voltage profiles compared to their non-incremental
+alternatives. The improvement was found to be starker for
+mutiphase feeders, wherein (non)-incremental rules do not
+reach the same equilibrium. Our findings motivate further
+research to possibly characterize the equilibria of control
+rules for multiphase feeders; the convergence of accelerated
+incremental rules for multiphase feeders; and to deal with
+chance-constrained formulations or ORD problems targeting
+phase imbalances.
+REFERENCES
+[1] IEEE Standard for Interconnection and Interoperability of DERs with
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+2280
+ 2340 2344
+106 116 119
+1574
+1577 1619
+Z
+296 372 650
+734
+841 933
+3155
+3156 3188
+3877
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+2844
+ 2940 2968
+2408
+2495 2500
+3552
+3741 3752
+2500
+2567 2764

0tAzT4oBgHgl3EQfevwc/content/tmp_files/load_file.txt

@@ -0,0 +1,504 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf,len=503
+page_content='1  Scalable Optimal Design of Incremental Volt/VAR  Control using Deep Neural Networks  Sarthak Gupta, Graduate Student Member, IEEE, Ali Mehrizi-Sani, Senior Member, IEEE,  Spyros Chatzivasileiadis, Senior Member, IEEE, and Vassilis Kekatos, Senior Member, IEEE  Abstract—Volt/VAR control rules facilitate the autonomous  operation of distributed energy resources (DER) to regulate  voltage in power distribution grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' According to non-incremental  control rules, such as the one mandated by the IEEE Standard  1547, the reactive power setpoint of each DER is computed as a  piecewise-linear curve of the local voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' However, the slopes  of such curves are upper-bounded to ensure stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' On the  other hand, incremental rules add a memory term into the  setpoint update, rendering them universally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' They can  thus attain enhanced steady-state voltage profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Optimal rule  design (ORD) for incremental rules can be formulated as a bilevel  program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' We put forth a scalable solution by reformulating ORD  as training a deep neural network (DNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' This DNN emulates the  Volt/VAR dynamics for incremental rules derived as iterations of  proximal gradient descent (PGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The rule parameters appear as  DNN weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' To reduce the DNN depth, we leverage Nesterov’s  accelerated PGD iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Analytical findings and numerical  tests corroborate that the proposed ORD solution can be neatly  adapted to single/multi-phase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Index Terms—IEEE Standard 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='8, incremental control  rules, multiphase feeders, proximal gradients, gradient backprop-  agation, deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' INTRODUCTION  Local Volt/VAR (Volt-Ampere Reactive) control facilitates  voltage regulation on distribution grids by providing reactive  power compensation from DERs equipped with smart invert-  ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Different from centralized control schemes which incur  large computational and communication burden, local rules  decide DER setpoints based on local measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Volt/VAR  control rules can be categorized into non-incremental and  incremental ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The former compute DER reactive power  setpoints based on local voltage readings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The IEEE Stan-  dard 1547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='8 prescribes such non-incremental control rules  as piecewise-linear functions of voltage [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' On the other  hand, incremental Volt/VAR rules compute the change in VAR  setpoints as a function of voltage [2]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The existing literature on designing Volt/VAR control rules  can be classified into stability- and optimality-centric works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Stability-centric works study the effect of Volt/VAR rules as a  closed-loop dynamical system, which may be rendered unsta-  ble under steep slopes of non-incremental rules [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' In fact,  to ensure stability, non-incremental rules may have to compro-  mise on the quality of their steady-state voltage profile [5],  [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Incremental rules however do not experience stability  limitations and can thus achieve improved voltage profiles  compared to their non-incremental counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Nonetheless,  such improvements may come at the expense of longer settling  times of the associated Volt/VAR dynamics [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Optimality-centric works focus on designing stable control  rules to minimize a voltage regulation objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' To this end,  optimization-based strategies have been employed to design  affine non-incremental rules using heuristics [9]–[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Two of  our recent works in [12] and [13] have addressed the problem  of optimally designing the slope, deadband, saturation, and  reference voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Reference [12] performs ORD via a bilevel  optimization applicable to single-phase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Reference [13]  proposes DNN-based digital twins that emulate Volt/VAR  dynamics, and reformulates ORD as a DNN training task for  single-/multi-phase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  This letter deals with optimally selecting the shape of  incremental Volt/VAR control rules, with contributions on  three fronts: c1) Although this optimal rule design (ORD)  task can be posed as a mixed-integer nonlinear optimization  program, it does not scale well with the numbers of DERs,  nodes, and grid loading scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' To address this challenge,  the genuine idea here is to reformulate ORD as a deep-  learning task and judiciously adapt the fast software modules  widely available for training deep neural networks (DNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  We have put forth a similar approach for designing non-  incremental control rules in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' However, migrating from  non-incremental to incremental rules is non-trivial due to the  different curve shapes, stability, and settling time properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  c2) To further expedite ORD for incremental rules, we suggest  implementing accelerated Nesterov-type variants of the rules  to yield a shallower DNN emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' c3) We also establish the  convergence of incremental rules on multiphase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Recently, reference [14] deals with the optimal design of  incremental rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' It uses DNNs with a single hidden layer  to model piecewise-linear functions and formulates ORD as  a reinforcement learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' While [14] also utilizes DNNs  to design incremental rules, we delineate from it in several  ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Reference [14] focuses on voltage control during tran-  sient dynamics, whereas this work aims at ORD to drive  steady-state voltages closer to unity and over different grid  loading scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Reference [14] utilizes a DNN to model  the piecewise-linear mapping of the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' In contrast, this  work develops a DNN-based digital twin that emulates end-  to-end Volt/VAR dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Lastly, we provide stability and  convergence analysis for single- and multiphase feeders alike,  whereas [14] applies only to single-phase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The rest of this letter is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Section II  models the feeder and discusses non-incremental and incre-  mental Volt/VAR control rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Section III formulates DNN-  based digital twins for Volt/VAR dynamics of incremental  rules, and their accelerated version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' It also presents ORD  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='01440v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='OC]  4 Jan 2023  2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Non-incremental Volt/VAR control rule provisioned by the IEEE Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  1547 for the interconnection of DERs [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  for single-phase feeders as a deep learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Section IV  extends the ORD process to multiphase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The incre-  mental rules are then benchmarked against non-incremental  rules from [13] using tests on real-world data, in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The letter is concluded in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' VOLT/VAR CONTROL RULES  Consider a radial feeder serving N buses equipped with  DERs, indexed by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Let (qℓ, q) collect reactive loads and  generations at all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Vectors (p, v) collect the net active  power injections and voltage magnitudes at all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The  impact of q on v can be approximately captured using the  linearized grid model [13]  v ≃ Xq + ˜v  (1)  where ˜v := Rp − Xqℓ + v01 models the underlying grid  conditions, and v0 is the substation voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Vector ˜v rep-  resents the impact of non-controlled quantities (p, qℓ) on  voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Matrices (R, X) depend on the feeder topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' For  single-phase feeders, they are symmetric positive definite with  positive entries [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' For multiphase feeders, they are non-  symmetric and have positive and negative entries [5], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Vector q in (1) carries the reactive injections by DERs  we would like to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Per the non-incremental rules of  the IEEE Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 1547 [1], DER setpoints are decided based  on the Volt/VAR curve of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 1, which is parameterized by  (¯v, δ, σ, ¯q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The standard further constrains these parameters  within a polytopic feasible set [1], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The negative slope of  the linear segment of the curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 1 can be expressed as  α :=  ¯q  σ − δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The interaction of Volt/VAR rules with the feeder gives  rise to nonlinear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' These dynamics are stable if  ∥ dg(α)X∥2 < 1, where dg(α) is a diagonal matrix carry-  ing the rule slopes over all buses on its diagonal [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The  equilibrium setpoints for DERs cannot be expressed in closed  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' However, they coincide with the minimizer of the convex  optimization problem [7]  min  −¯q≤q≤¯q  1  2q⊤Xq+q⊤(˜v−¯v)+ 1  2q⊤ dg−1(α)q+δ⊤|q| (2)  where |q| applies the absolute value on q entrywise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Prob-  lem (2) depends on rule parameters (¯v, δ, α, ¯q) across all buses,  collected in the 4N-long vector z := (¯v, δ, α, ¯q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' We denote  by qz(˜v) the equilibrium setpoints, and by  vz(˜v) = Xqz(˜v) + ˜v  (3)  the related equilibrium voltages reached by Volt/VAR rules  parameterized by z under grid conditions ˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Optimal rule design (ORD) can be stated as the task of  selecting z to bring equilibrium voltages vz(˜v) close to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  To cater to diverse conditions, the utility may sample loading  scenarios {˜vs}S  s=1 for the next hour, and find z as  z∗ ∈ arg min  z  F(z) := 1  S  S  �  s=1  ∥vz(˜vs) − 1∥2  2  (ORD)  subject to (3) and z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Once found, the customized rules z∗ are sent to DERs to  operate autonomously over the next hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Note that vz(˜vs)  depends on z because the equilibrium setpoints qz(vs) in (3)  are the minimizers of problem (2), which is parameterized by  z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' When solving (ORD) for non-incremental rules, the feasible  set Z consists of the polytopic constraints imposed on z by  the IEEE Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 1547 as well as additional constraints on α  to ensure ∥ dg(α)X∥2 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' see [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Therefore, the feasible  set Z can be quite confined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' This can lead to less desirable  voltage profiles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' that is, higher objective values F(z∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The aforesaid issue can be addressed by replacing the non-  incremental Volt/VAR rules of IEEE Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 1547 by incremental  ones as suggested in [2]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Incremental rules express the  change rather than the actual value in setpoints as a function  of voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' One option for incremental rules is to implement  a proximal gradient descent (PGD) algorithm solving (2)  as proposed in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' In this case, the control rule coincides  with the PGD iterations, which are implemented by DERs  in a decentralized fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Using incremental rules, set Z is  enlarged as now we only need to ensure  z ≥ 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='95 · 1 ≤ ¯v ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='05 · 1  and that ¯q are within the reactive power ratings of the DERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The PGD algorithm is an extension of gradient descent to  handle constraints and non-differentiable costs [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' At iteration  t, PGD proceeds with two steps: s1) It first computes the gradi-  ent of the first two terms of F(z), that is Xqt+˜v−¯v = vt−¯v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Here qt is the latest estimate of the minimizer of (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' s2) PGD  then updates qt+1 as the minimizer of  min  −¯q≤q≤¯q  1  2q⊤ dg−1(α)q + δ⊤|q| + 1  2µ∥q − (vt − ¯v)∥2  2 (4)  for a step size µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The last problem involves the last two  terms in the cost of (2) regularized by the Euclidean distance  of q to the gradient (vt − ¯v) computed in step s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Converting PGD to control rules, step s1) is performed by  the physics of the feeder when injecting qt and measuring  the local voltage deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Step s2) is run by each DER  independently as (4) is separable across buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Using the  subdifferential, solving (4) provides the update [5]  yt  n = ˜αn ·  �  qt  n − µ(vt  n − ¯vn)  �  (5a)  fq  UA3  qt+1  n  = gn  �  yt  n  �  (5b)  where gn(yn) is the proximal operator  gn(yn) :=  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  +¯qn  , yn > qn + µ˜δn  yn − µ˜δn  , µ˜δn < yn ≤ qn + µ˜δn  0  , − µ˜δn ≤ yn ≤ µ˜δn  yn + µ˜δn  , − qn − µ˜δn ≤ yn < −µ˜δn  −qn  , yn < −qn − µ˜δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  (6)  and the new parameters (˜αn, ˜δn) are defined as  ˜αn :=  1  1 + µ/αn  and  ˜δn :=  δn  1 + µ/αn  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The proximal operator is plotted in the top panel of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Note that in (5), rule parameters are transformed from repre-  sentation z = (¯v, δ, α, ¯q) to representation ˜z := (¯v, ˜δ, ˜α, ¯q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  This is without loss of generality as the transformation is a  bijection, and so one can work exclusively with ˜z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The feasible  set ˜Z for ˜z is similar to Z with the addition that ˜α ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' As  with non-incremental rules, the rules in (6) are driven by local  data, but now qt+1  n  depends on (vt  n, qt  n), and not vt  n alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Both  types of rules solve (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Hence, they both converge to the same  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The advantage of incremental rules is that they are  stable for all α as long as µ < 2/λmax(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' It is worth  stressing that z does not have the same physical interpretation  as in non-incremental rules (slopes, deadband, or saturation),  though z parameterizes (2) for both rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Accelerated incremental rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Although PGD rules en-  large Z, their settling times can be long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' They reach an  ε-optimal cost of (2) within − 2 log ε  log 2 κ (X) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Here  κ(X) := λmax(X)/λmin(X) is the condition number of  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' References [5], [16] put forth accelerated incremental  rules based on accelerated PGD (APGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' These rules need  − 2 log ε  log 2  �  κ (X) iterations to attain an ε-optimal cost, and take  the form  ˜yt  n := (1 + βt) yt  n − βtyt−1  n  (7a)  qt+1  n  := gn  �  ˜yt  n  �  (7b)  where βt := t−1  t+2, while yt  n and gn(yn) are as defined in (5a)  and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Updates (7) remain local, but introduce additional  memory as qt+1  n  depends on (vt  n, qt  n) and (vt−1  n  , qt−1  n  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' DEEP LEARNING FOR OPTIMAL RULE DESIGN (ORD)  IN SINGLE-PHASE FEEDERS  Solving (ORD) is challenging as it is a nonconvex bilevel  program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Although it can be modeled as a mixed-integer  nonlinear program, such an approach does not scale well with  the number of DERs and/or scenarios for non-incremental  rules [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Seeking a more scalable solution, we reformulate  (ORD) as a deep learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The key idea is to design  a DNN that emulates Volt/VAR dynamics under the control  rule of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' To this end, note that gn(yn) is a piecewise-  linear function with four breakpoints [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Interestingly, this  operator can be expressed as the superposition of four rectified  linear units (ReLUs) as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 2, where ReLUs are  denoted by ρ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The intercepts of the ReLUs depend linearly  on (˜δn, ¯qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Proximal operator g(y) expressed as a sum of four shifted rectified  linear units (ReLUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  A DNN emulating the accelerated incremental rules of (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Plain  incremental rules can be modeled by dropping the second layer (setting βt =  0) and ignoring output yt  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Building on this, one APGD iteration for DER n can be  implemented by the 4-layer DNN in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 3, whose weights  depend affinely on (¯vn, ˜δn, ˜αn, ¯qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' This DNN takes (qt  n, vt  n)  as its input, and computes (qt+1  n  , yt  n) at its output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' It is  termed ICn and will be used as a building block to emulate  Volt/VAR dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' This is accomplished by the recursive  neural network (RNN) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Here blocks ICn are  arranged vertically to model the parallel operation of DERs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Their outputs qt+1 are multiplied by X, and the new voltage  is computed as vt+1 = Xqt+1 + ˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' This is repeated T times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Thanks to the RNN structure, there is weight sharing, so the  number of DNN weights is 4N rather than 4NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The RNN takes a grid loading vector ˜vs as its input, the rule  parameters ˜z as weights, and computes the voltages vT  ˜z (˜vs)  at time T at its output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' For the output vT  ˜z (˜vs) to approximate  well equilibrium voltages, the depth T can be chosen by the  convergence rate of PGD as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' For the DNN of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 4 to ensure ∥Φ (˜v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' z) −  v∗(z)∥2 ≤ ϵ1 ∀ ˜v, its depth T should satisfy  T ≥  �κ − 1  2  �  log  �2∥X∥2∥ˆq∥2  ϵ1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  (8)  +qH  b+  qp  uon  一  qn  t  p  t  p  uon  p4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Recurrent neural network (RNN) implementation for accelerated  incremental Volt/VAR control rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Proof: From the control rule of (5b), it follows that  ∥qt − q∗∥2 = ∥g  �  yt�  − g (y∗) ∥2  ≤ ∥yt − y∗∥2  = ∥ dg(˜α)(I − µX)  �  qt−1 − q∗�  ∥2  ≤ ∥ dg(˜α)∥2 · ∥I − µX∥2 · ∥qt−1 − q∗∥2  ≤ ∥I − µX∥2 · ∥qt−1 − q∗∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  (9)  The first inequality stems from the non-expansive property of  the proximal operator g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The next equality follows from (5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The second inequality from the sub-multiplicative property of  the spectral norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The last inequality follows by the definition  of spectral norm and because ˜αn ≤ 1 for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  If ∥I − µX∥2 < 1, inequality (9) implies that the dynamics  in (5) are a non-expansive mapping, and thus, are stable and  converge to q∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Condition ∥I − µX∥2 < 1 holds when µ <  2/λmax(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The norm ∥I − µX∥2 achieves its minimum of  �  1 −  2  κ+1  �  when  µ0 :=  2  λmax(X) + λmin(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Plugging µ0 into (9) and unfolding the dynamics over t  provides  ∥qt − q∗∥2 ≤  �  1 −  2  κ+1  �t  ∥q0 − q∗∥2  ≤ 2  �  1 −  2  κ+1  �t  ∥ˆq∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  For  the  voltage  approximation  error  ∥vT − v∗∥2  =  ∥X  �  qT − q∗�  ∥2 at time T to be smaller than ϵ1, we need  ∥vT − v∗∥2 ≤ 2∥X∥2 · ∥ˆq∥2 ·  �  1 −  2  κ + 1  �T  ≤ ϵ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  This can be achieved by selecting T such that  T ≥  log  �  2∥X∥2∥ˆq∥2  ϵ1  �  log  �  1 +  2  κ−1  �  ≥  �κ − 1  2  �  log 2∥X∥2∥ˆq∥2  ϵ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  where the last inequality follows from log(1 + x) ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Plugging the values ∥X∥2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='463 and κ = 848 for the  IEEE 37-bus feeder, ∥ˆq∥2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='1, and ϵ1 = 10−5 in (8), yields  T ≥ 2, 892 layers, which is relatively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' A key contributor  to this large T is the κ term in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' This promulgates the  adoption of accelerated rules (7), which are known to have  O(√κ) dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Interestingly, during implementation, one  does not need to fix T to the above worst-case bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Leveraging dynamic computation graphs offered by Python  libraries such as Pytorch, one may determine T ‘on the  fly’ depending on the convergence of vt between pairs of  successive layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Since the RNN emulates Volt/VAR dynamics, it can surro-  gate vz(˜vs) in (ORD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Then (ORD) can be posed as training  a DNN over its weights ˜z ∈  ˜Z or z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Grid loading  scenarios {˜vs}S  s=1 are treated as features and equilibrium  voltages vz(˜vs) as predictions that should be brought close to  the target value of 1 for scenarios s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The DNN can be trained  using stochastic projected gradient descent (SPGD) as [13]  ˜zi+1 =  �  ˜zi − λ  2B ∇˜zi  � �  s∈Bi  ∥Φ(˜vs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' ˜z) − 1∥2  2  ��  ˜  Z  (10)  where λ > 0 is the learning rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' set Bi is a batch of  B scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' and [·] ˜  Z is the projection onto  ˜Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Since  ˜Z  consists of simple box constraints, projection essentially means  clipping the values to the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Lastly ∇˜zi(·) represents the  gradient with respect to ˜z evaluated at ˜z = ˜zi, and is  calculated efficiently thanks to gradient back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Although our DNN-based ORD assumed PGD-based rule, it  may be applicable to other incremental rules too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  A discussion about control rules and their DNN-based  emulators is due.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Recall that all three types of Volt/VAR  control rules (non-incremental, incremental, and accelerated  incremental) reach the same equilibrium voltages, if stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The emulators aim at computing these equilibrium voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  A natural question is whether the DNN emulator could im-  plement a rule of a type different from the rule actually  implemented on the feeder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' This may be desirable to leverage  the advantages of two types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Some caution is needed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' If  the feeder implements non-incremental rules, but incremental  rules converge faster to equilibrium voltages, it makes sense  for the emulator to implement incremental rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Of course,  in this case, stability constraints on the non-incremental rules  have to be enforced during DNN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The reverse is not  recommended: If the emulator implements non-incremental  rules, its parameters z should be constrained to be stable and  that would be a restriction of the actual ORD problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Finally,  given the convergence advantage of accelerated incremental  rules, they are always preferable over plain incremental rules  for the DNN implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' This showcases the utility of  accelerated control rules even if they are not actually imple-  mented on the feeder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' DEEP LEARNING FOR OPTIMAL RULE DESIGN (ORD)  IN MULTIPHASE FEEDERS  In multiphase feeders, matrix X is non-symmetric and has  both positive and negative entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Therefore, the rule analysis  and design of Section III has to be revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' For example,  equilibrium setpoints cannot be found as the minimizers of  IC1  七  q  t+1  q  IC2  t t  t-1  IC N5  an optimization problem as with (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Moreover, increasing q  does not mean that all voltages increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  In multiphase feeders, the non-incremental rules of IEEE  Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 1547 remain stable as long as ∥ dg(α)X∥2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' This is  the same condition as in the single-phase setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' How about  the stability and equilibrium of incremental rules in multiphase  feeders?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Recall that for single-phase rules, incremental rules  were obtained as the PGD iterations solving (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Lacking an  equivalent inner optimization for multiphase feeders precludes  a similar approach here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Despite the incremental rules of (5)  do not correspond to PGD iterates anymore, they can still be  shown to be stable for multiphase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Let UΛU⊤ be the eigen-decomposition of  matrix XX⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The incremental rules of (5) are stable for  multiphase feeders if their step size is selected as µ <  λmin  �  Λ−1/2U⊤ �  X + X⊤�  UΛ−1/2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The claim follows readily by adopting the proof of Proposi-  tion 1: If µ is selected as above, then ∥I − µX∥2 < 1 follows  from [5, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Similar to the single-phase case, incremental  rules in multiphase feeders allow us to enlarge the feasible  set Z of rule parameters z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' It is worth stressing that different  from the single-phase setting, incremental and non-incremental  rules do not converge to the same equilibrium on multiphase  feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The ORD task for multiphase feeders can also be formulated  as a deep-learning task, with some modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Firstly,  matrices R and X need to be altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Secondly, the DNNs  for multiphase feeders have 12N trainable parameters, since  each layer consists of 3N building modules corresponding  to bus/phase (node) combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Lastly, the step size has  to be selected per Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Adopting the proof of  Proposition 1, we next find the minimum DNN depth in  multiphase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Let the DNN of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 4 implement the incre-  mental rules of (5) on multiphase feeders with µ selected per  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The DNN depth T ensuring voltage approxi-  mation error ∥Φ (˜v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' z) − v∗(z)∥2 ≤ ϵ1 is  T ≥  log  ϵ1  2∥X∥2∥ˆq∥2  log ∥I − µX∥2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  We next numerically evaluate the proposed DNN-based  ORD approach in single- and multiphase feeders, and contrast  the performance of incremental control rules with that of non-  incremental rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' NUMERICAL TESTS  We benchmark the performance of DNN-based incremental  rules against non-incremental rules from [13] on single- and  multiphase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Real-world data were sourced from the  Smart* project on April 2, 2011 [17], as explained in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The DNNs were implemented and trained using Pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  We first compare (non)-incremental rules, both designed  via DNN training for the single-phase IEEE 37-bus feeder  of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Homes with IDs 20–369 were averaged 10  at a time and successively added as active loads to buses  2–26 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Active generation from solar  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The IEEE 37-bus feeder converted to single-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Node num-  bering follows the format node number {panel ID}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' DERs at buses  {6, 9, 11, 12, 15, 16, 20, 22, 24, 25} provide reactive power control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' the rest  operate at unit power factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  TABLE I  INCREMENTAL VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' NON-INCREMENTAL VOLT/VAR CONTROL RULES ON  THE SINGLE-PHASE IEEE 37-BUS FEEDER  Time  q = 0  Non-incremental  Incremental  Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=')  Time (s)  Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=')  Time (s)  Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=')  1 pm  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='01 · 10−3  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='98  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='68 · 10−4  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='39  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='66 · 10−4  2 pm  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='13 · 10−3  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='93  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='26 · 10−4  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='91  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='25 · 10−4  3 pm  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='24 · 10−3  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='02  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='59 · 10−4  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='97  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='50 · 10−4  4 pm  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='12 · 10−3  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='47 · 10−4  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='48 · 10−4  5 pm  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='53 · 10−4  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='37  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='70 · 10−5  374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='01  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='90 · 10−5  panels was also added, as per the mapping in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Buses {6, 9, 11, 12, 15, 16, 20, 22, 24, 25} were assumed to  host DERs with Volt/VAR control customized per bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Incremental rules were simulated in their accelerated ren-  dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Both sets of rules were trained over S = 80 scenarios  and 200 epochs with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='001, using the  Adam optimizer, and setting µ = 1 for incremental rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' To  ensure repeatability, the results were repeated across several  time periods between 1–6 PM, and are compiled in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Incremental rules obtained marginally lower objectives than  non-incremental rules across all periods, with a somewhat  significant difference for the 5 PM period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' This behavior is  explained because incremental rules allow for a larger set Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  DNN-based incremental control rules were also contrasted  with their non-incremental ones on the multiphase IEEE 13-  bus feeder, using the testing setup from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Active loads were  sampled 10 at a time from homes with IDs 20-379 and added  to all three phases for the buses 1-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Figure 6 also shows the  solar panel assignments shown in Fig 6 for solar generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  Lastly, nine DERs with inverters were added across phases  and bus indices as shown in Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  The learning rates for non-incremental and incremental  DNNs were set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='001, respectively, with the design  Z  Z  Z  Z  Z  Z6  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Multiphase IEEE 13-bus distribution feeder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  TABLE II  INCREMENTAL VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' NON-INCREMENTAL VOLT/VAR CONTROL RULES ON  THE MULTIPHASE IEEE 13-BUS FEEDER  Time  q = 0  Non-incremental  Incremental  Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=')  Time (s)  Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=')  Time (s)  Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=')  1 pm  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='51 · 10−3  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='15 · 10−3  199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
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+page_content='94 · 10−4  263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='37  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='16 · 10−4  4 pm  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='03 · 10−4  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
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+page_content='26 · 10−4  126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
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+page_content='47 · 10−4  5 pm  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='51 · 10−4  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='58  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='11 · 10−4  129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='71  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='95 · 10−4  parameters z := (¯v, δ, σ, α) initialized to feasible values  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='95, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Table V compares the performance of  the two rule categories over multiple periods for S = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  While incremental rules took longer times to train, they were  successful in lowering the cost F(z) by more than 50%, thus  yielding improved voltage profiles across all periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' CONCLUSIONS  We have devised a DNN approach to optimally design  incremental Volt/VAR control rules for single- and multi-phase  feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The key idea is to construct a DNN that emulates  end-to-end the associated Volt/VAR dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The DNN takes  grid conditions as the input, the rule parameters as weights,  and outputs the associated equilibrium voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Leveraging  the convergence rates of the related optimization algorithms,  we have provided bounds on the minimum depth of the  DNN emulator to approximate equilibrium voltages within the  desired accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' We have also established the stability of  incremental control rules for multiphase feeders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Numerical  tests have demonstrated that the designed control rules attain  improved voltage profiles compared to their non-incremental  alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' The improvement was found to be starker for  mutiphase feeders, wherein (non)-incremental rules do not  reach the same equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' Our findings motivate further  research to possibly characterize the equilibria of control  rules for multiphase feeders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' the convergence of accelerated  incremental rules for multiphase feeders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
+page_content=' and to deal with  chance-constrained formulations or ORD problems targeting  phase imbalances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAzT4oBgHgl3EQfevwc/content/2301.01440v1.pdf'}
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+Single-point spin Chern number in a supercell framework
+Roberta Favata1 and Antimo Marrazzo1, ∗
+1Dipartimento di Fisica, Universit`a di Trieste, Strada Costiera 11, I-34151 Trieste, Italy
+(Dated: January 9, 2023)
+We present an approach for the calculation of the Z2 topological invariant in non-crystalline
+two-dimensional quantum spin Hall insulators. While topological invariants were originally mathe-
+matically introduced for crystalline periodic systems, and crucially hinge on tracking the evolution
+of occupied states through the Brillouin zone, the introduction of disorder or dynamical effects can
+break the translational symmetry and imply the use of larger simulation cells, where the k−point
+sampling is typically reduced to the single Γ-point. Here, we introduce a single-point formula for
+the spin Chern number that enables to adopt the supercell framework, where a single Hamiltonian
+diagonalisation is performed. Inspired by the work of E. Prodan [Phys. Rev. B, 80, 12 (2009)], our
+single-point approach allows to calculate the spin Chern number even when the spin operator ˆsz
+does not commute with the Hamiltonian, as in the presence of Rashba spin-orbit coupling. We val-
+idate our method on the Kane-Mele model, both pristine and in the presence of Anderson disorder.
+Finally, we investigate the disorder-driven transition from the trivial phase to the topological state
+known as topological Anderson insulator. Beyond disordered systems, our approach is particularly
+useful to investigate the role of defects, to study topological alloys and in the context of ab-initio
+molecular dynamics simulations at finite temperature.
+I.
+INTRODUCTION
+Two-dimensional (2D) topological insulators (TI) are
+materials with an insulating bulk and robust edge states
+protected by the non-trivial topology of the bulk elec-
+tronic structure [1, 2].
+These systems are discussed
+through topological invariants, integer quantities which
+characterise the ground-state wavefunction in the bulk.
+As long as the topological invariant is non-trivial and,
+possibly, the symmetries needed to define that topology
+are preserved, the material is said to be in a topolog-
+ical phase. These invariants are geometrical properties
+of the electronic structure, as they are defined in terms
+of quantities such as the Berry phase or the Berry cur-
+vature, which involve derivatives of the occupied states
+in reciprocal space with respect to the quasi-momentum
+k [2]. Standard geometrical formulas are usually discre-
+tised on a regular mesh of k-points for numerical imple-
+mentation. However, most electronic structure calcula-
+tions for non-crystalline systems are normally performed
+by diagonalising the Hamiltonian at a single k-point in a
+large supercell. Usually the Γ point at the center of the
+Brillouin zone (BZ) is considered, although potentially
+more efficient choices based on the Baldereschi point [3]
+can be employed. The derivation of single-point formu-
+las for geometrical and topological properties is not at all
+a trivial task, although successful single-point formalism
+have been developed for the Berry phase [4], the orbital
+magnetisation and the Chern number [5].
+In this work, we target the calculation of the topologi-
+cal invariant for non-crystalline 2D insulators with time-
+reversal (TR) symmetry. For these systems, encompass-
+ing all non-magnetic 2D materials [6], the invariant ν is
+∗ antimo.marrazzo@units.it
+a Z2 number: if ν = 0 the topology is trivial, otherwise
+if ν = 1 we have a quantum spin Hall insulator (QSHI),
+where topologically-protected gapless helical edge states
+cross the bulk gap [2].
+Over the years, several meth-
+ods have been developed to calculate the Z2 invariant
+in crystalline systems with periodic boundary conditions
+(PBCs).
+In the following, we briefly outline some of the most
+popular and practical methods in the context of elec-
+tronic structure simulations.
+If inversion symmetry is
+present, there is a particularly simple method introduced
+by Fu and Kane [7], which requires the knowledge of the
+parity of the occupied states at the four TR-invariant
+points in the BZ. In the more general case, the Z2 in-
+variant can be obtained by tracking the evolution of
+hermaphrodite [8] (a.k.a. hybrid) Wannier charge cen-
+tres [9–11], or equivalently the eigenvalues of the Wilson
+loop [12–14], over half BZ. More recently, a generalisa-
+tion of the Fu-Kane approach based on elementary band
+representations [15, 16] has allowed to calculate the in-
+variant by using only the knowledge of the irreducible
+representations of the occupied states at selected high-
+symmetry points in the BZ [16]. The Z2 invariant can
+be also computed as an individual Chern number [2] on
+half of the Hilbert space [10, 11], where the split is per-
+formed by two projectors which are smooth and related
+by TR symmetry. Although several formulas to compute
+the Z2 invariant have been introduced, all the ones we
+mentioned, and most other existing approaches, require
+the knowledge of the occupied states at multiple k-points
+and become ill-defined for non-crystalline systems; hence
+in the supercell framework they are of no avail.
+Nonetheless, a number of methods have been proposed
+to deal with non-periodic systems. Some of these [17–
+19] calculate the Z2 invariant by means of a Pfaffian
+with twisted boundary conditions, as firstly advocated
+by Kane and Mele in their original discussion of the Z2
+arXiv:2301.02612v1  [cond-mat.mes-hall]  6 Jan 2023
+
+2
+invariant in presence of disorder and electron-electron in-
+teractions [20]. A different method is based on construct-
+ing the Z2 invariant from the scattering matrix of the
+system at the Fermi level [21, 22]. Further, there exists
+a formulation based on the non-commutative index the-
+orem [23, 24], where the Z2 index for disordered topolog-
+ical insulators is computed from the discrete spectrum
+of a certain compact operator, which is defined as the
+difference of a proper pair of projection operators [25–
+27].
+An alternative non-commmutative approach was
+proposed by Loring and Hastings [28, 29] and relates the
+Z2 index to the topological obstruction to approximat-
+ing almost commuting matrices by exactly commuting
+matrices; its robustness with respect to the introduction
+of disorder has been investigated in Ref. [30]. The most
+practical approach from the point of electronic structure
+simulations has been arguably put forward by Huang
+and Liu [31, 32], who addressed the problem of calcu-
+lating the Z2 invariant for non-periodic system in the
+context of quantum spin Hall quasicrystals, and intro-
+duced the spin Bott index, which measures the commu-
+tativity of the projected position operators. The connec-
+tion between the Bott indices and Chern or Z2 invariants
+has been investigated theoretically [28–30, 33], while nu-
+merical simulations [32, 34, 35] provided evidence that
+Bott indices can be used to study non-periodic topolog-
+ical systems. Still, it is conceptually rather unsatisfac-
+tory that the calculation of topological invariants in a
+supercell framework requires introducing radically differ-
+ent formalisms, which call for rather non-trivial equiva-
+lence proofs and extensive testing. As a matter of fact,
+the use of the primitive cell and k-points is an arbitrary—
+although indeed very convenient—choice; there is no con-
+ceptual reason preventing bona fide Z2 invariants to be
+calculated directly in the supercell by deriving a suitable
+single-point limit. In addition, it is important to assess
+the convergence with respect to the system size, as dif-
+ferent approaches might deliver the same correct answer
+at very different computational costs. For instance, re-
+cent works [32, 33] claimed that the difference between
+the Chern number and Bott index is within a correc-
+tion of the order O(1/L), where L is the linear size of
+the system. Such slow convergence can hinder the study
+of the system close to a topological phase transition; in
+fact Huang and Liu empirically added a singular value
+decomposition (SVD) to their algorithm to improve an
+otherwise slow convergence [32].
+Here, we take a different approach, that essentially
+combines the work of Ceresoli and Resta on the single-
+point Chern number [5] and the insights from Prodan
+on a generalised spin Chern number [36]. Notably, our
+single-point invariant is directly derived by its parent for-
+mula for crystalline systems, it shows exponential conver-
+gence with the supercell size, both in the pristine and dis-
+ordered case, it is easy to implement in electronic struc-
+ture codes, and it works well also in presence of strong
+Rashba spin-orbit coupling (SOC).
+II.
+METHODS
+In absence of spin-mixing spin-orbit interactions, the
+spin operator ˆsz commutes with the Hamiltonian and it
+is possible to discuss the Z2 invariant in terms of the spin
+Chern number. In this case, the occupied states diago-
+nalise ˆsz and can be divided in two subsets, either purely
+spin-up or spin-down, and the regular Chern number can
+be calculated for each spin. As soon as the Hamiltonian
+does not commute any more with ˆsz, for instance because
+Rashba SOC is present, such simple-minded spin Chern
+number cannot be defined any more. Notably, Prodan
+has shown [36] that it is possible to generalise this def-
+inition by projecting the spin operator on the occupied
+states:
+Pz = P(k)ˆszP(k),
+(1)
+where P is the ground-state projector
+P(k) =
+�
+n
+|unk⟩ ⟨unk| ,
+(2)
+unk are the periodic part of the Bloch eigenstates and
+n labels the occupied state at each k-point in the BZ.
+Then, we diagonalise Pz:
+Pz |uλ⟩ = sλ |uλ⟩ .
+(3)
+If only diagonal SOC terms are present, the eigenvalue
+spectrum of Pz consists of two values only sλ = ± 1
+2 and
+one can select a single spin component by choosing the
+eigenstates which correspond to one of the two eigenval-
+ues sλ.
+The crucial observation made by Prodan [36]
+is that, even if Rashba SOC is present, the spectrum of
+Pz displays two separate bands of eigenvalues symmet-
+ric around the origin and one can still introduce a well-
+defined spin Chern number by selecting the eigenvectors
+with positive (or negative) eigenvalues. Finally, the spin
+Chern number can be computed as:
+Cs = C+ − C−
+2
+mod 2
+(4)
+where C± are calculated on the uλ eigenstates with posi-
+tive and negative eigenvalues respectively; in general it is
+sufficient to compute either C+ or C− only and consider
+its parity. The results are of paramount practical rele-
+vance, as it is typically much simpler to deal with a for-
+mulation based on generalised Chern numbers, which can
+be written as full BZ integrals and do not require taking
+into account TR symmetry or complex gauge fixing, as
+required instead by more general Z2 formulations [20, 37].
+In principle, if the Rashba interaction is strong enough
+then the gap of the Pz spectrum might close, preventing
+the spin Chern number to be defined. Remarkably, as
+we will discuss in full detail in the Sec. III, this does not
+seem to occur in practice. As long as the system is in-
+sulating, Eq. 4 is well defined even if the Rashba SOC is
+several times larger than the diagonal SOC. Hence, we
+
+3
+adopt the approach of Prodan [36] and target the deriva-
+tion of a single-point formula. In order to obtain the cor-
+rect single-point limit, we follow the approach of Ceresoli
+and Resta [5] for the derivation of the single-point Chern
+number in TR-broken systems (the latter admit a Z topo-
+logical invariant). Let us start with the formula for the
+generalised spin Chern number in 2D periodic systems:
+Cσ = 1
+2π
+�
+BZ
+TrσΩxy(k)dk
+= − 1
+π
+�
+sλ=σ
+�
+BZ
+Im ⟨∂kxuλ(k)|∂kyuλ(k)⟩ dkxdky,
+(5)
+where uλ are the eigenvectors of Pz (see Eq. 3) and σ = ±
+corresponds to one of the sectors of the Pz spectrum.
+Now we consider the parallelogram Brillouin zone and
+change coordinate system to have a rectangular integra-
+tion domain:
+Cσ = − 1
+π Im
+�
+sλ=σ
+� b1
+0
+dk1
+� b2
+0
+dk2 ⟨∂k1uλ(k)|∂k2uλ(k)⟩
+≃ −|b1||b2|
+π
+Im
+�
+sλ=σ
+⟨∂k1uλ(k)|∂k2uλ(k)⟩ |k=Γ,
+(6)
+where b1,2 are the two reciprocal lattice vectors and the
+last step is performed in the limit of a very large supercell.
+In the same limit, we can calculate derivatives through
+finite differences:
+∂kj |uλ(k)⟩ |k=Γ = lim
+η→0
+|uλ(ηbj)⟩ − |uλ(Γ)⟩
+η|bj|
+,
+(7)
+where we can drop the limit for a large supercell and
+just consider the difference |uλ(bj)⟩ − |uλ(Γ)⟩. Eq. 7 re-
+quires a differentiable function, which is not guaranteed
+in numerical diagonalisations. Hence, we fix the gauge
+by adopting a discretised version of the covariant deriva-
+tive [38, 39] as successfully performed for the Chern num-
+ber by Ceresoli and Resta [5]. One replaces the states
+with their “duals”:
+|˜un(bj)⟩ =
+�
+m
+S−1
+mn(bj) |um(bj)⟩
+(8)
+where
+we
+define
+the
+overlap
+matrix
+Snm(bj)
+=
+⟨un(Γ)|um(bj)⟩
+and
+the
+dual
+states
+satisfy
+⟨un(Γ)|˜um(bj)⟩
+=
+δnm.
+Next,
+we construct the
+states un(bj) by imposing the periodic gauge, which
+allows us to perform a single diagonalisation at Γ:
+|uλ(bj)⟩ = e−ibj·r |uλ(Γ)⟩ .
+(9)
+The states in Eq. 9 are Hamiltonian eigenstates, but they
+might correspond to a different eigenvalue with respect
+to the one at Γ; the ordering is anyway fixed by the
+covariant derivative. We note in passing, that while a
+non-trivial Chern number would prevent the adoption of
+a periodic gauge for the wavefunction, here the periodic
+gauge is only temporarily imposed to build each |un(bj)⟩
+from the knowledge of the |un(Γ)⟩, but it is effectively
+replaced by the parallel transport gauge enforced by the
+covariant derivative. The final single-point formula for
+the spin Chern number is
+C(asym)
+σ
+= −|b1||b2|
+π
+Im
+�
+sλ=σ
+⟨˜uλ(b1)|˜uλ(b2)⟩ .
+(10)
+In Eq. 10, we emphasise with the superscript “asym” the
+implicit choice made in Eq. 7, which corresponds to the
+right-hand derivative. In fact, an alternative choice is the
+symmetric derivative
+∂kj |uλ(k)⟩ |k=Γ ≃ |uλ(bj)⟩ − |uλ(−bj)⟩
+2|bj|
+,
+(11)
+which can also be computed with a single Γ-only diago-
+nalisation and leads to the following formula for the spin
+Chern number:
+C(sym)
+σ
+= −|b1||b2|
+4π
+Im
+�
+sλ=σ
+(⟨˜uλ(b1)| − ⟨˜uλ(−b1)|) (|˜uλ(b2)⟩ − |˜uλ(−b2)⟩) .
+(12)
+In Sec. III, we will show how the symmetric formula con-
+verges much faster than the asymmetric version, at es-
+sentially the same computational cost.
+We
+have
+implemented
+the
+single-point
+formulas
+in a dedicated Python package, freely available on
+GitHub [40].
+The code provides user-friendly inter-
+faces to two popular tight-binding packages such as
+PythTB [41] and TBmodels [42], and it can be easily
+interfaced to other codes.
+III.
+NUMERICAL RESULTS AND DISCUSSION
+We validate our approach on the paradigmatic Kane-
+Mele (KM) model [20, 43] on the honeycomb lattice, both
+pristine and in presence of Anderson disorder (see Fig. 1).
+
+4
+FIG. 1. The Kane-Mele model in the supercell approach. Left panel: pristine Kane-Mele model, the primitive cell is shown
+in orange while a 3 × 3 supercell is marked in blue. Right panel: random realisation of a disordered Kane-Mele model in a
+3 × 3 supercell (green) with periodic boundary conditions, where different colours are used to represent the on-site terms. In
+the following, supercells are labelled by their integer size L × L (in units of the pristine primitive cell) and the corresponding
+number of sites N = 2L2.
+The tight-binding Hamiltonian reads
+HKM = t
+�
+⟨i,j⟩
+c†
+icj + ∆
+�
+i
+ξic†
+ici
++ iλSO
+�
+⟨⟨i,j⟩⟩
+νijc†
+iσzcj
+(13)
++ iλR
+�
+⟨i,j⟩
+c†
+i(σ × ˆdij)zcj,
+where i and j run over all sites in the lattice and the
+creation and annihilation operators are expressed in the
+contracted form c†
+i = (c†
+i↑, c†
+i↓). The first term is a real
+nearest-neighbour hopping (denoted by ⟨ , ⟩), if taken
+alone that would yield four (pair-degenerate) bands with
+gapless Dirac cones centred on the high-symmetry points
+K and K
+′ in the Brillouin zone.
+The second term
+is a staggered on-site potential (ξi = ±1 is the sub-
+lattice index of the i−th site) while the third term is
+the KM SOC [20, 43] which involves a complex next-
+nearest neighbour hopping (denoted by ⟨⟨ , ⟩⟩) with a
+spin-dependent amplitude proportional to the Pauli ma-
+trix σz.
+The factor νij = sign(d1 × d2)z depends on
+the orientation of the vectors d1 and d2 along the two
+bonds connecting i to the next-nearest neighbour site j.
+The fourth term is the Rashba SOC and is a complex
+nearest-neighbour hopping with off-diagonal spin com-
+ponents, where σ = (σx, σy, σz) is the vector of Pauli
+matrices and ˆdij is the unit vector between sites j and
+i. In the following, we consider a KM Hamiltonian at
+fixed parameters t = 1 and λSO = 0.03 t, which ensure
+that the energy gap is insulating all over the entire phase
+diagram [20, 43].
+A.
+Validation and convergence tests for crystalline
+systems
+In the single-point approach, the topological invari-
+ants become exact integer numbers only in the thermody-
+namic limit of an infinite supercell. First, we test the con-
+vergence properties of the single-point spin Chern num-
+ber (SPSCN) on the pristine KM model, in both asym-
+metric (Eq. 10) and symmetric (Eq. 12) formulation. We
+inspect the SPSCN as a function of the supercell size L,
+here defined as the number of primitive cells along each
+lattice vector that makes the supercell L×L (see Fig. 1);
+the number of sites inside the supercell is N = 2L2. A
+representation of a supercell 3 × 3 is given in the left-
+hand panel of Fig. 1. In our calculations only values of L
+which are multiple of 3 are considered, to always include
+the special points K and K
+′ folded at Γ.
+We bench-
+mark the accuracy of the formulas inside the Z2-even
+and Z2-odd domains in Fig. 2.
+The symmetric for-
+mula converges faster than the asymmetric one in both
+trivial and topological phases. Remarkably, the quantity
+∆Cσ = |Cσ(L)−Cσ(∞)|, which is the difference between
+the spin Chern number given by the single-point formulas
+at finite sizes and the exact value obtained in the ther-
+modynamic limit, decreases exponentially in both formu-
+lations. However, the global prefactor in the symmetric
+case is an order of magnitude smaller than the one of the
+asymmetric formula, leading to more accurate results at
+significantly smaller sizes L. Hence, in the following we
+adopt the symmetric formula only and study the topo-
+logical phase transition as a function of the on-site ∆,
+results are reported in Fig. 3.
+Our SPSCN is able to
+reproduce the sharp topological transition already at rel-
+atively small supercell sizes, as shown in the left-hand
+panel of Fig. 3. The band gap vanishes on the boundary
+of the phase transition and in the corresponding neigh-
+bourhood of parameters convergence is slower and larger
+
+5
+6
+12
+18
+24
+30
+36
+42
+48
+54
+60
+L
+−0.100
+−0.075
+−0.050
+−0.025
+0.000
+0.025
+0.050
+0.075
+0.100
+Single-point Cσ
+Asymmetric
+Symmetric
+Exact value
+0
+20
+40
+60
+L
+10−4
+10−3
+10−2
+10−1
+∆Cσ
+−5
+0
+5
+∆/λSO
+−5
+0
+5
+λR/λSO
+6
+12
+18
+24
+30
+36
+42
+48
+54
+60
+L
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+1.05
+1.10
+1.15
+Single-point Cσ
+Asymmetric
+Symmetric
+Exact value
+0
+20
+40
+60
+L
+10−4
+10−3
+10−2
+10−1
+∆Cσ
+−5
+0
+5
+∆/λSO
+−5
+0
+5
+λR/λSO
+FIG. 2. Convergence of the single-point spin Chern number, in its symmetric and asymmetric implementation, with respect
+to supercell size for the Kane-Mele model, where the Hamiltonian is diagonalised at the Γ-point only. In the uppest insets, a
+sketch of the corresponding point in the pristine phase diagram. The lowest insets show the difference between the single-point
+calculations of the spin Chern number and the thermodynamic limit. Left panel: the spin Chern number converges to zero in
+the trivial phase (∆/λSO = 5.5, λR/λSO = 3). Right panel: in the topological phase (∆/λSO = 0.8 , λR/λSO = 2) the spin
+Chern number converges to one. In all cases, the asymptotic convergence is exponential, but the symmetric formula converges
+visibly faster than its asymmetric counterpart.
+0
+1
+2
+3
+4
+5
+6
+∆/λSO
+−0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Single-point Cσ
+L = 9
+L = 24
+L = 51
+Exact value
+−5
+0
+5
+∆/λSO
+−5
+0
+5
+λR/λSO
+0
+1
+2
+3
+4
+5
+6
+∆/λSO
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+˜Eg
+L = 9
+L = 24
+L = 51
+FIG. 3. Left panel: the single-point spin Chern number (symmetric formula) versus the on-site term ∆ at fixed λR/λSO = 2
+for the Kane-Mele model. Different supercell sizes are considered (L = 9, 21, 51, and corresponding number of sites N =
+162, 882, 5202). As the supercell size increases, the transition becomes sharper and approaches the analytical solution. Right
+panel: gap ˜Eg of the P ˆszP operator versus the on-site term ∆ for the same supercell sizes as on the left-hand panel.
+A
+non-vanishing ˜Eg guarantees that the spin Chern number is well defined.
+supercell sizes must be employed. In the right-hand panel
+of Fig. 3, we show how the gap ˜Eg of the Pz operator
+varies across the topological phase transition, but always
+remains finite, ensuring that our single-point invariant is
+everywhere well defined. Then, we validate the SPSCN
+by calculating the entire topological phase diagram of the
+KM model, which is reported in the upper panel of Fig. 4.
+Notably, the method can distinguish topological and triv-
+ial phases even for small, but still finite, values of both
+the gap of the Hamiltonian and the gap of Pz (lower left-
+hand panel in Fig. 4). Larger differences between the SP-
+SCN and the exact value (zero), which are visibile in the
+upper-left side of the topological phase diagram (marked
+in blue), are finite size effects and are reduced for large
+supercells, as highlighted in the lower right-hand panel in
+Fig. 4: in that region both to Hamiltonian and Pz oper-
+ators gap are indeed very small. Therefore, our formulas
+works well also in presence of very strong Rashba SOC
+and small band gaps.
+
+6
+0
+1
+2
+3
+4
+5
+6
+∆/λSO
+0
+1
+2
+3
+4
+λR/λSO
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+Single-point Cσ
+0
+1
+2
+3
+4
+5
+6
+∆/λSO
+0
+1
+2
+3
+4
+λR/λSO
+1e-14
+0.2
+0.4
+0.6
+0.8
+1
+˜Eg
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+λR/λSO
+−0.4
+−0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Single-point Cσ
+L = 21
+L = 36
+L = 48
+Exact value
+−5
+0
+5
+∆/λSO
+−5
+0
+5
+λR/λSO
+FIG. 4. Upper panel: topological phase diagram of the Kane-Mele model calculated with the single-point spin Chern number
+(symmetric formula), for a supercell size L = 36 containing N = 2592 sites. Black dashed line marks the analytical solution
+for the semi-metallic state separating the topological and trivial phases. Lower-left panel: gap ˜Eg of the P ˆszP operator for
+the same calculations performed in the upper panel. Notably, ˜Eg is non-vanishing over all the phase diagram and guarantees
+that the spin Chern number is well defined everywhere. Lower-right panel: the single-point spin Chern number versus the
+Rashba coupling λR, at fixed ∆/λSO = 0.3, for different supercell size L = 21, 36, 48 and corresponding number of sites
+N = 882, 2592, 4608. In that region of the phase diagram, band gaps are very small and finite size effects intensify; still the
+single-point approach can distinguish the two phases.
+B.
+Disorder-driven topological phase transitions
+The presence of disorder is often modelled by means
+of an ensemble of large supercells, each representing a
+specific random realisation as schematically represented
+in the right-hand panel of Fig. 1. In electronic structure
+simulations, defect calculations are performed by consid-
+ering large supercells, to suppress the spurious interac-
+tions due to the periodic replicas. Alloys are often sim-
+ulated through the so-called special quasi-random struc-
+tures [44]. In addition, a non-perturbative treatment of
+temperature effects always require working with super-
+cells, being a single structure with special atomic dis-
+placements [45] or a collection of snapshots obtained from
+ab initio molecular dynamics.
+The SPSCN particularly suits this framework, and we
+now assess the accuracy and convergence properties of
+our formula on the KM model supplemented by an An-
+derson disorder term [46], where we highlight its capa-
+bility to detect disorder-driven topological transitions.
+We emphasise that the simple KM model in presence
+of rather strong Anderson disorder is used as a proto-
+type and a proxy for testing, although our approach
+targets the more general scenario mentioned above, of
+supercell calculations, either for model Hamiltonians or
+first-principles simulations.
+The Hamiltonian of the disordered KM model reads
+Hdis = HKM +
+�
+i
+wic†
+ici,
+(14)
+where wi ∈
+�
+− W
+2 , W
+2
+�
+is a randomly distributed on-site
+potential and W is the disorder strength which, in the fol-
+lowing, is reported in units of the nearest-neighbour hop-
+ping amplitude t. In Fig. 5 we test the convergence of the
+single-point formulas (Eqs. 10 and 12) with increasing su-
+percell size L for the disorder strength W/t = 1, which is
+weak enough not to destroy the topological phases of the
+corresponding pristine KM model. The SPSCN is eval-
+
+7
+5
+10
+15
+20
+25
+30
+35
+40
+45
+L
+−0.15
+−0.10
+−0.05
+0.00
+0.05
+0.10
+0.15
+Single-point Cσ
+Asymmetric
+Symmetric
+0
+20
+40
+L
+10−4
+10−3
+10−2
+10−1
+∆Cσ
+−5
+0
+5
+∆/λSO
+−5
+0
+5
+λR/λSO
+5
+10
+15
+20
+25
+30
+35
+40
+45
+L
+0.75
+0.80
+0.85
+0.90
+0.95
+1.00
+1.05
+1.10
+1.15
+Single-point Cσ
+Asymmetric
+Symmetric
+0
+20
+40
+L
+10−4
+10−3
+10−2
+10−1
+∆Cσ
+−5
+0
+5
+∆/λSO
+−5
+0
+5
+λR/λSO
+FIG. 5. Convergence of the single-point spin Chern number, in its symmetric and asymmetric implementation, with respect to
+supercell size L for the disordered Kane-Mele model. We report the average and standard deviation of the single-point invariant
+calculated on M = 100 realisations with disorder strength W/t = 1. In the upper insets, the point in the corresponding pristine
+phase diagram is shown. In the lowest insets, we report the difference between the mean value and the thermodynamic limit as
+a function of L. Left panel: the spin Chern number converges to zero for ∆/λSO = 5.5 and λR/λSO = 3. Right panel: the spin
+Chern number converges to one for ∆/λSO = 0.8 and λR/λSO = 2. Also in presence of disorder, the asymptotic convergence
+is exponential and the symmetric formula converges visibly faster than its asymmetric counterpart. Statistical fluctuations are
+very small and negligible at almost any supercell size.
+uated as the mean value over M realisations of random
+disorder with supercells of size L × L. Also in presence
+of disorder, the convergence of the formulas is exponen-
+tial and the symmetric version converges faster than the
+asymmetric one. In addition, we consider increasing dis-
+order strengths and study the robustness of the topolog-
+ical phase, results are reported in Fig. 6. For sufficiently
+strong disorder, the topological phase is destroyed and
+the SPSCN becomes trivial. As expected, the width of
+the phase transition becomes smaller with increasing su-
+percell sizes. As investigated in Ref. [47], for a certain
+range of parameters, the disordered KM model given by
+Eq. 14 displays a topological state called topological An-
+derson insulator (TAI). It is a phase of quantized con-
+ductance which is obtained adding Anderson disorder to
+a trivial insulator or metal which are relatively close to
+a topological phase transition [48–50]. The mechanism
+for this disorder-induced transition has been discussed in
+terms of a renormalization of the model parameters such
+as the on-site term [49].
+The weak-disorder boundary
+of a TAI can be studied within an effective-medium the-
+ory and the self-consistent Born approximation [47, 49],
+but these perturbative approach might fail in the strong-
+disorder regime, where the TAI phase is destroyed in
+favour of a trivial insulating phase, as we show next.
+In Fig. 7 we use the SPSCN to inspect these topological
+phase transitions driven by disorder. In order to compare
+with previous work on the disordered KM model [47] and
+for the sake of clarity, we consider a value of λSO = 0.3 t
+which is an order of magnitude greater than the one used
+for the previous examples. First, we fix λR = 0 (left-
+hand panel) and observe that the TAI appears at about
+W/t = 2, in agreement with the conductance calcula-
+tion in [47] and the spin Bott index results in [32] (note
+the factor of two with respect to the W defined therein).
+Then, we consider finite Rashba SOC and show the re-
+sults in the right-hand panel of Fig. 7, where we note
+that the TAI region has become narrower, in agreement
+with Ref. [47]. A check on the gap ˜Eg of operator Pz
+is performed for every SPSCN calculation in presence of
+disorder: Anderson disorder never fully closes the gap
+and the invariant can always be computed.
+IV.
+CONCLUSIONS
+In this work, we have introduced a robust and effi-
+cient single-point formula to calculate the Z2 topological
+invariant in non-crystalline 2D materials. We have val-
+idated our method with supercell numerical simulations
+on the KM model, both pristine and disordered.
+Our
+approach can reproduce the entire phase diagram of the
+KM model, where each calculation requires only a single-
+point diagonalisation in the supercell framework, even in
+presence of strong Rashba SOC. In addition, we have
+extensively tested our method in presence of Anderson
+disorder, and we have shown how the single-point for-
+mula can correctly describe disorder-driven topological
+phase transitions. In particular, we have discussed both
+the process where disorder destroys the topological phase
+and where disorder actually promotes it, as for the TAI
+phase; that is in agreement with calculations of the con-
+ductance [22, 47] and spin Bott index [32] reported in
+the literature. Our single-point approach converges ex-
+ponentially with size, so it is typically sufficient to work
+with relatively small supercells, which is critical for ap-
+
+8
+0
+2
+4
+6
+8
+10
+W/t
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+Single-point Cσ
+L = 15
+L = 42
+−5
+0
+5
+∆/λSO
+−5
+0
+5
+λR/λSO
+0
+2
+4
+6
+8
+10
+W/t
+10−1
+100
+min( ˜Eg)
+FIG. 6.
+Robustness of the topological phase with respect to disorder.
+The symmetric single-point spin Chern number is
+calculated as function of disorder strength W/t, starting from the system in the topological phase (∆/λSO = 3, λR/λSO = 1).
+For each W, we report the mean and standard deviation over M = 50 realisations of Anderson disorder for supercells of sizes
+L = 15, 42 and number of sites N = 450, 3528 respectively. Upper inset: a sketch of the point where the calculations are
+computed reported on the pristine phase diagram (W/t = 0). Lower inset: minimum value, over the disorder realizations, of
+the gap ˜Eg of the P ˆszP operator as a function of W/t. With increasing supercell size L, the transition becomes sharper. ˜Eg
+does not vanish with Anderson disorder and the approach performs well also in the strong-disorder regime.
+0
+2
+4
+6
+8
+10
+12
+W/t
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+Single-point Cσ
+L = 15
+L = 42
+−5
+0
+5
+∆/λSO
+−5
+0
+5
+λR/λSO
+0
+2
+4
+6
+8
+10
+12
+W/t
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+Single-point Cσ
+L = 15
+L = 42
+−5
+0
+5
+∆/λSO
+−5
+0
+5
+λR/λSO
+0
+2
+4
+6
+8
+10
+12
+W/t
+10−3
+10−2
+10−1
+100
+min( ˜Eg)
+FIG. 7. Topological Anderson insulator (TAI). The symmetric single-point spin Chern number is calculated as function of
+disorder strength W/t, starting from the system in a trivial state close to the phase transition. We report the mean value and
+the standard deviation of single-point invariant over M = 50 realisations of Anderson disorder for supercells of sizes L = 15, 42
+and corresponding numbers of sites N = 450, 3528 respectively. For 5 ≤ W/t ≤ 10 the number of random realisations is
+purposely increased to M = 100 to reduce the standard deviation.
+Left panel: TAI state in absence of Rashba coupling
+(∆/λSO = 5.5, λR = 0). Right panel: TAI at finite Rashba coupling (∆/λSO = 5.3, λR/λSO = 1). Here, the minimum value
+of the gap ˜Eg (over M disorder realizations) is reported versus W/t in the lower inset (the same plot is not present in the
+right-hand left panel since ˜Eg is constantly equal to one for λR = 0).
+
+9
+plications in ab initio modelling. One of the side benefits
+of adopting Prodan’s approach is that the formula can,
+at least in principle, be meaningful also in presence of
+weak TR-breaking perturbations [36]. This feature could
+be useful to study how the bulk topology is affected by
+the presence of magnetic impurities, or of a magnetic
+substrate through the proximity effect; even though the
+absence of TR symmetry would allow backscattering be-
+tween the two helical edge states. To encourage the use
+of our approach, we release a dedicated Python package
+that allows to seamlessly calculate the single-point Chern
+(Z) and spin-Chern (Z2) invariants of any TB model
+thanks to dedicated interfaces to PythTB and TBmodels,
+two very popular TB codes. Notably, these two packages
+also allow working with Wannier Hamiltonians, which
+are read in the format produced by Wannier90 [51, 52];
+that provides a simple way to apply our work in the con-
+text of first-principles calculations.
+Then, it would be
+interesting to explore the effect of the TB approximation
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+and it could be implemented with limited effort directly
+into plane-wave first-principles codes, such as Quantum
+ESPRESSO [53, 54]. In short, our approach allows study-
+ing 2D topological insulators in a supercell framework,
+which is crucial to investigate very relevant phenomena
+such as disorder, defects, alloying, and to study dynam-
+ical and temperature effects through ab initio molecular
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+

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+ 
+Physarum Inspired Bicycle Lane Network Design 
+in a Congested Mega City  
+ 
+ 
+By 
+Md. Ahsan Habib 
+Roll: 1507082 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Department of Computer Science and Engineering 
+Khulna University of Engineering & Technology 
+Khulna 9203, Bangladesh 
+ 
+ 
+March 2020
+
+ 
+ 
+ii 
+ 
+Certification 
+ 
+ 
+The thesis titled “Physarum Inspired Bicycle Lane Network Design in a Congested 
+Mega City” submitted by Md. Ahsan Habib, Roll No: 1507082, Academic Year: 2018-
+19, for partial fulfillment of the requirements for the degree of “Bachelor of Science in 
+Computer Science and Engineering”. 
+ 
+Supervisor 
+ 
+   
+                                                                
+          Dr. Muhammad Aminul Haque Akhand 
+ 
+Professor 
+  
+Dept. of Computer Science and Engineering  
+ 
+Khulna University of Engineering & Technology 
+ 
+Khulna, Bangladesh. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+iii 
+ 
+ 
+Acknowledgements 
+ 
+First and foremost, I must sense grateful to and wish to acknowledge my insightful 
+indebtedness to Dr. Muhammad Aminul Haque Akhand, Professor of Department of 
+Computer Science and Engineering and the supervisor of the thesis. His unfathomable 
+knowledge in this field influenced me to carry out this thesis up to this point. His endless 
+endurance, scholarly guidance, continual encouragement, 
+constant and lively 
+supervision, constructive criticism, priceless suggestion made it possible to come up to 
+this phase. Without his inspiring, enthusiasm and encouragement, this work could not 
+be completed. 
+Last, but by no means least, I thank Allah for the talents and abilities I was given that 
+made it possible to undertake this thesis. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+iv 
+ 
+ 
+Abstract 
+ 
+Mobility is a key factor in urban life and transport network plays a vital role in mobility. 
+Worse transport network having less mobility is one of the key reasons to decline the 
+living standard in any unplanned mega city. Transport mobility enhancement in an 
+unplanned mega city is always challenging due to various constraints including 
+complex design and high cost involvement. The aim of this thesis is to enhance 
+transport mobility in a megacity introducing a bicycle lane. To design the bicycle lane 
+natural Physarum, brainless single celled multi-nucleated protist, is studied and 
+modified for better optimization. Recently Physarum inspired techniques are drawn 
+significant attention to the construction of effective networks. Exiting Physarum 
+inspired models effectively and efficiently solves different problems including 
+transport network design and modification and implication for bicycle lane is the unique 
+contribution of this study. Central area of Dhaka, the capital city of Bangladesh, is 
+considered to analyze and design the bicycle lane network bypassing primary roads. 
+ 
+
+ 
+ 
+v 
+ 
+ 
+Contents 
+ 
+    Title Page   
+ 
+ 
+ 
+ 
+ 
+ 
+     i 
+    Certification 
+ 
+ 
+ 
+ 
+ 
+ 
+    ii 
+    Acknowledgements  
+ 
+ 
+ 
+ 
+ 
+   iii 
+    Abstract 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+   iv 
+    Contents  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+    v 
+    List of Tables  
+ 
+ 
+ 
+ 
+ 
+ 
+  vii 
+    List of Figures 
+ 
+ 
+ 
+ 
+ 
+             viii 
+CHAPTER 1      Introduction 
+1 
+ 
+ 
+    1.1 Overview of Transport Network in a Mega City  
+ 
+    1 
+ 
+ 
+    1.2 Motivation 
+ 
+ 
+ 
+ 
+ 
+ 
+    1 
+                            1.3 Objectives of the Thesis 
+2 
+                            1.4 Organization of the Thesis 
+3 
+CHAPTER 2      Physarum Inspired Network Design  
+4 
+                             2.1 Physarum and its Properties 
+4 
+                             2.2 Network Design Inspired on Physarum 
+5 
+                             2.3 Review of Existing Physarum Inspired Works 
+6 
+ 
+ 
+ 
+2.3.1 Transpiration Network Design                                          6 
+ 
+ 
+ 
+2.3.1 Other Optimization Task 
+ 
+ 
+ 
+    7 
+CHAPTER 3      Physarum Inspired Bicycle Lane Design in an Unplanned  
+ 
+ 
+ Mega City                                                                                         10 
+                          3.1 Mobility Problem in an Unplanned Mega City: Dhaka as a Case 
+Study                                                                                                             9 
+ 
+ 
+ 
+3.1.1 History and Overview of Dhaka City 
+ 
+ 
+    9 
+ 
+ 
+ 
+3.1.2 Transportation Crisis in Dhaka City 
+ 
+ 
+    9 
+ 
+ 
+ 
+3.1.3 Effect of Transportation Crisis to Other Problems 
+  13 
+                            3.2 Importance of Bicycle Lane in an Mega City                                     14 
+                            3.3 Challenges to Increase Mobility in Dhaka City 
+15 
+                            3.4 Bicycle Lane Design in an Unplanned Mega City 
+16 
+                            3.5 Significance of Study 
+23 
+CHAPTER 4      Experimental Studies 
+24 
+                            4.1 Experimental Settings 
+24 
+
+ 
+ 
+vi 
+ 
+                            4.2 Bicycle Lane Network Design in a Prominent Area 
+Error! 
+Bookmark not defined. 
+                                                  4.2.1 Network Design 
+25 
+                                  4.2.2 Effectiveness Analysis                                                  32 
+                                          4.2.2.1 Time Saving 
+33 
+                                          4.2.2.2 Fuel and Cost Saving 
+33 
+                                          4.2.2.3 CO2 Emission Reduction 
+36 
+                            4.3 Bicycle Lane Network Design for Entire Dhaka City 
+36 
+                                  4.3.1 Network Design                                                             36 
+                                  4.3.2 Effectiveness Analysis                                                  38                                          
+                                          4.3.2.1 Time Saving 
+39 
+                                          4.3.2.2 Fuel and Cost Saving 
+42 
+                                          4.3.2.3 CO2 emission reduction 
+44 
+CHAPTER 5      Conclusions 
+46 
+ 
+ 
+     5.1 Achievements  
+ 
+ 
+ 
+ 
+ 
+  46 
+ 
+ 
+     5.2 Future Study 
+ 
+ 
+ 
+ 
+ 
+ 
+  46 
+References 
+47 
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+vii 
+ 
+ 
+List of Tables 
+ 
+ 
+ 
+ 
+Table No 
+Description 
+Page 
+2.1 
+Network construction using Physarum. 
+7 
+4.1 
+The locations of prominent area of Dhaka city including 
+traffic pressure. 
+27 
+4.2 
+Routes from node point 1. 
+32 
+4.3 
+Time comparison in car, bus and bicycle considering 
+from node point 1. 
+34 
+4.4 
+Time saving per day. 
+34 
+4.5 
+Cost comparison in car, bus and bicycle considering 
+from node point 1. 
+35 
+4.6 
+Cost saving per day. 
+35 
+4.7 
+CO2 emission reduction per day. 
+36 
+4.8 
+The general data on locations in Dhaka city, including 
+population, area, and traffic pressure. 
+38 
+4.9 
+Routes from node point 7. 
+39 
+4.10 
+Time comparison in car, bus and bicycle considering 
+from node point 7. 
+41 
+4.11 
+Time saving per day. 
+41 
+4.12 
+Cost comparison in car, bus and bicycle considering from 
+node point 1. 
+43 
+4.13 
+Cost saving per day. 
+43 
+4.14 
+CO2 emission reduction per day. 
+44 
+
+ 
+ 
+viii 
+ 
+ 
+List of Figures 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure No 
+Description 
+Page 
+2.1 
+Physarum polycephalum. 
+4 
+3.1 
+Farmgate to University of Dhaka routes and time 
+needed in driving mode. 
+11 
+3.2 
+Physarum inspired network design of 11 nodes. 
+16 
+3.3 
+Real traffic network. 
+17 
+4.1. 
+Selected Dhaka city Map. 
+19 
+4.2 
+Network design using modified Physarum inspired 
+technique. 
+21 
+4.3 
+Dhaka city Map. 
+28 
+4.4 
+Constructed network 29 points of Dhaka city. 
+ 
+29 
+
+ 
+ 
+2 
+ 
+CHAPTER 1 
+Introduction 
+A transport network can be described as a collection of linear features that permits 
+either vehicular movement or flow of some commodity. The characteristics of 
+Physarum can be used to design a transportation network. This chapter discusses the 
+background study of network design, objectives, and organization of the thesis.  
+1.1 Overview of Transport Network in a Mega City 
+Mobility, a key factor in planning and designing urban transport, is a fundamental part 
+of human beings [1].   For mobility purposes, people can use both motorized and non-
+motorized vehicles within a city. Motorized vehicles like buses, cars, motorbikes, 
+cycles, etc. are hazardous in many kinds [2]. Non-motorized transports involve walking 
+and cycling as well as variants such as small-wheeled transportation like skates, 
+skateboards, push scooters and hand carts [3]. Nowadays, non-motorized mobility is 
+trendy [4].  
+ 
+The idea of mega city emerged to characterize the world's largest metropolitan 
+agglomerations at the end of the 20th century. In the 1970s, only two mega cities had 
+over ten million residents. Currently, 9.9% of the urban population globally resides in 
+23 megacities. It is projected that in 2025, the number will increase to 37 if 13.6% of 
+global urban population are to be accommodated [5].  
+ 
+A transportation network is the formation of a spatial network that enables vehicle 
+movement or the flow of some commodities. A vast network of rail, subways and bus 
+lines passes through well-organized mega cities. There are also exist special footways 
+and networks for cycling routes. 
+1.2 Motivation 
+In a well-planned and well-organized mega city, both footways and roads are available 
+for mobility purpose but in case of an unplanned and unorganized mega city, both 
+footpaths and roads are hardly available. In some cases, footpaths are snatched by 
+
+ 
+ 
+3 
+ 
+hawkers and street vendors. The public transport system is defined by far a lack of 
+people's desired travel needs in terms of mobility, reliability, convenience, pace and 
+safety. In fact, some transports like buses are considered unreliable and time consuming 
+to reach their destinations. The Texas Transportation Institute reported a delay of 3.6 
+billion vehicle-hours in the 75 biggest metropolitan regions in 2000, culminating in 5.7 
+billion U.S. gallons (21.6 billion liters) of waste fuel and a loss of productivity of $67.5 
+billion or around 0.7 percent of GDP.  
+ 
+Bicycling or walking activity may help increase blood flow, release endorphins, and 
+decrease overall stress. It can even help to improve mental health and energy by 
+tracking 30 minutes of  bicycling or walking a day[6]–[9]. Efficient and effective 
+bicycle lane network design in an unplanned and unorganized mega-city can minimize 
+total travel time, fuel usage, costs, carbon dioxide (CO2) emission, etc. Physarum 
+polycephalum is multi-headed, brainless, a giant multi-nucleated, single-celled protist 
+that can solve different complex problems. Physarum networks are believed to have 
+achieved a good balance between cost, efficiency, and resilience. 
+1.3 Objectives of the Thesis 
+In the case of an unorganized and unplanned mega city, where the transportation 
+network is congested and unplanned and there are hardly any footpaths available and 
+no further transport facilities can be expanded. So there are some huge problems in 
+those cities like traffic jam, noise pollution, air pollution, CO2 emission, etc. With a 
+planned lane network with non-motorized vehicles nearly all of this problem can be 
+solved. Since it is not feasible to completely rebuild the transportation network and 
+infrastructure of a mega city but possible to transform mega city towards a green city. 
+The objective of this study is given below: 
+ Study of Physarum 
+ Physarum related paper study 
+ Network design 
+ Bicycle lane network in mega city  
+
+ 
+ 
+4 
+ 
+1.4 Organization of the Thesis 
+The main attraction of this thesis is to present a modified Physarum inspired technique 
+to construct bicycle lane network design. The thesis has five chapters. An introduction 
+to network design and Physarum has been given in Chapter 1. Chapter-wise overviews 
+of the rest of the thesis are as follows:  
+Chapter 2: Describes the literature review that includes a brief description of Physarum 
+with its properties and previous related work to Physarum inspired network design. 
+Chapter 3: Explains the proposed modified Physarum Inspired Bicycle Lane Design 
+in an Unplanned Mega City in detail. 
+Chapter 4: Reports the experimental result of modified Physarum Inspired Bicycle 
+Lane Design. Also, in this chapter, a case study of Dhaka is demonstrated. Finally, 
+Chapter 5: This chapter is for the conclusions of this thesis together with the outline 
+of future directions of research opened by this work.
+
+ 
+ 
+ 
+CHAPTER 2 
+Physarum Inspired Network Design  
+Physarum polycephalum is a brainless amoeboid organism. Physarum-inspired network 
+design model has demonstrated extraordinary skill in designing effective networks. In 
+this chapter firstly, we discuss the Physarum polycephalum, secondly the Physarum-
+based network design and lastly, the existing network design inspired by Physarum. 
+2.1 Physarum and its Properties 
+Physarum polycephalum, accurately the 'many-headed' slime mold, is a gigantic multi-
+nucleated but single-celled protist [10]. The slime mold Physarum polycephalum 
+creates a form of spatial memory by avoiding areas it has previously explored to 
+navigate in a complex environment[11]. Recently, Physarum polycephalum (true slime 
+mold) has arisen as a fascinating illustration of biological computation through 
+morphogenesis[12]. Although it is a single-cell organism, studies have shown that the 
+Physarum can overcome different minimum cost flow problems through its growth 
+process[12]. In the following Fig. 2.1, an example of the Physarum polycephalum is 
+shown.
+ 
+Figure 2.1: Physarum polycephalum. [61]. 
+ 
+
+FS
+FS
+FS
+Source
+FS
+FS 
+ 
+5 
+ 
+Here Physarum polycephalum is shown to grow up the network towards the FSs from 
+the source. (FS = Food Source) 
+2.2 Network Design Inspired on Physarum  
+The intelligent behavior of slime mold was first observed by Nakagaki et al. in 
+2000[13]. In previous biological experiments, Physarum-inspired network model has 
+exhibited an extraordinary intelligence to build efficient networks to connect multiple 
+food sources. Physarum networks are believed to have achieved a good balance 
+between cost, efficiency, and resilience. For instance, Physarum constructed networks 
+with comparable qualities to those of the Tokyo rail system in a renowned experiment 
+performed by Tero et al. in 2010[14]. 
+They developed a mathematical model for adaptive network construction to emulate 
+the behavior of Physarum which is based on feedback loops between the thickness of 
+each tube and internal protoplasmic flow in which high rates of streaming stimulate an 
+increase in tube diameter, whereas tubes tend to decline at low flow rates. The edges 
+represent plasmodial tubes in which protoplasm flows, and nodes are junctions between 
+tubes. They consider the pressure at nodes 𝑖 and 𝑗 are 𝑃𝑖 and 𝑃𝑗, respectively, and the 
+two nodes are connected by a cylinder of length 𝐿𝑖𝑗 and radius 𝑟𝑖𝑗. They assume that 
+the flow is laminar and follows the Hagen-Poiseuille equation, the flux through the tube 
+is, 
+𝑄𝑖𝑗 =  𝜋𝑟𝑖𝑗
+4(𝑃𝑖 − 𝑃𝑗)
+8𝜀𝐿𝑖𝑗
+  =  𝐷𝑖𝑗(𝑃𝑖 − 𝑃𝑗)
+𝐿𝑖𝑗
+  ,                                               (2.1) 
+here 𝜀 is the viscosity of the fluid, and 𝐷𝑖𝑗 =  
+𝜋𝑟𝑖𝑗
+4
+8𝜀   is a measure of the conductivity of 
+the tube. As the length 𝐿𝑖𝑗 is a constant, the behavior of the network is described by the 
+conductivities of the edges.  
+The constrains must be maintained, 
+ 
+  ∑ 𝑄1𝑗 = 𝐼0
+𝑗
+,  
+For source node-1 
+ 
+  ∑ 𝑄2𝑗 = −𝐼0
+𝑗
+, 
+For sink node-2 
+ 
+  ∑ 𝑄𝑖𝑗 = 0
+𝑗
+,  
+Inflow and outflow must be conserved 
+
+ 
+ 
+6 
+ 
+To accommodate the adaptive behavior of the plasmodium, the conductivity of each 
+tube evolves according to 
+𝑑𝐷𝑖𝑗
+𝑑𝑡 = 𝑓(|𝑄𝑖𝑗|) − 𝐷𝑖𝑗, where 𝑓(|𝑄𝑖𝑗|) describes the 
+expansion of tubes in response to the flux and 𝐷𝑖𝑗 represents the rate of tube 
+constriction, so the tubes will gradually disappear. 
+The functional 𝑓(|𝑄|) = 
+|𝑄|𝛾
+(1+ |𝑄|𝛾) which describes a sigmoidal response where 𝛾 is a 
+parameter that controls the nonlinearity of feedback (𝛾 > 0).  
+2.3 Review of Existing Physarum Inspired Works  
+Physarum can successfully overcome many problems in real life even more 
+complicated problems. In this section initially, we discuss and summarize about the 
+Physarum inspired network design techniques, and then discuss about other 
+optimization problems solved using Physarum inspired methods. 
+2.3.1 Transpiration Network Design  
+To link several food points Physarum can build high-quality networks. A mathematical 
+model of the adaptive dynamics of a transport network of the true slime mold that shows 
+path-finding behavior in a maze is developed in 2007 by Tero et al. [15]. In 2010, 
+Physarum developed networks shows similar qualities to the Tokyo rail system in a 
+famous experiment conducted by Tero et al. [14]. Since then, Physarum inspired other 
+real-world transport networks, such as Iberian motorways [16] and Mexican Federal 
+highways [17] have also been constructed. Adamatzky et al. [18] develops a model to 
+construct networks on major urban areas of China. Becker et al. [19] developed in 2011 
+a fault tolerant connection networks for the Tokyo rail system using an agent based 
+simulation of Physarum polycephalum. Physarum-inspired cellular automaton (CA)-
+based network designing model was developed by Tsompanas et al. [20] inspired by 
+Slime Mould. Zhang et al. [21] recently proposed a method to solve the problem of 
+network design in supply chain for multiple source nodes and multiple sink nodes.  
+Physarum is excellent at doing other network design [22].Here we summarize various 
+works of network construction using Physarum inspired technique in the following 
+Table 2.1. 
+ 
+
+ 
+ 
+7 
+ 
+ 
+2.3.2 Other Optimization Task 
+Nakagaki et al. in 2000 [13] observed that Physarum productively found the shortest 
+path between two selected points in a maze. In addition, the Physarum can solve many 
+other famous problems like the shortest paths [23]–[25], towers of Hanoi problem [26] 
+and minimum risk problem [27]. Physarum can effectively solve many other complex 
+problems in the real world like traveling salesman problem [28]–[30], population 
+migration [31], etc. Logic gates design and boolean operations can be performed by a 
+slime mold network [32], [33]. Chaining these logic gates together can enable a slime 
+mold computer to perform binary computation operations. Physarum works very well 
+in logical computing as well[34]–[38]. Identifying critical components [39], [40] and 
+many other problems [41], [42] are effectively and efficiently solved through Physarum 
+bio-inspired technique. Most interestingly, many other studies have shown that 
+Table 2.1: Network construction using Physarum. 
+Authors & Year 
+Title of Paper 
+Contribution 
+Tero et al., 2007 
+[15] 
+A mathematical model for 
+adaptive transport network in 
+path finding by true slime mold 
+Model for adaptive transport 
+network in Path-finding in a maze 
+Tero et al., 2010 
+[14] 
+Rules for Biologically Inspired 
+Adaptive Network Design 
+Tokyo Rail Network construction 
+Adamatzky et al., 
+2011 [16] 
+Rebuilding Iberian motorways 
+with slime mould 
+Iberian motorway network 
+construction 
+Adamatzky et al., 
+2011 [17] 
+Approximating Mexican 
+highways with slime mould 
+Mexican Federal highway 
+network construction 
+Adamatzky et al., 
+2013 [18] 
+Slime mould imitates transport 
+networks in China 
+Slime mould protoplasmic 
+networks on major urban areas of 
+China 
+Becker et al., 2011 
+[19] 
+Design of fault tolerant 
+networks with agent-based 
+simulation of Physarum 
+polycephalum 
+Construction of fault tolerant 
+connection networks for the 
+Tokyo rail system using an agent 
+based simulation of Physarum 
+polycephalum 
+Tsompanas et al., 
+2015 [20] 
+Evolving Transport Networks 
+With Cellular Automata Models 
+Inspired by Slime Mould 
+Physarum-inspired cellular 
+automaton (CA)-based network 
+designing model 
+Zhang et al. 2016 
+[21] 
+A Physarum-inspired approach 
+to supply chain network design 
+Supply chain network design 
+ 
+ 
+
+ 
+ 
+8 
+ 
+Physarum's tubular topologies often mimic those of complex mathematical networks 
+[43], [44] like the Steiner tree problems [45]–[49].  
+ 
+Studies with Physarum related works showed that the organism can solve many 
+complex real-life problems efficiently and effectively, particularly in the sense of 
+network design. This can be applied with some changes for designing the bicycle lane 
+network. In this work bicycle lane network is planned using local lanes in congested 
+mega city. 
+ 
+ 
+  
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+10 
+ 
+CHAPTER 3 
+Physarum Inspired Bicycle Lane Design in an Unplanned  
+Mega City 
+An unplanned mega city suffers various problems including transporation and mobility. 
+Transport mobility enhancement in an unplanned mega city is always challenging due 
+to various constraints including complex design and high cost involvement. In this 
+thesis, we try to increase the mobility in an unplanned mega city Dhaka. In this chapter, 
+problems of an unplanned mega city Dhaka are addressed firstly, then challenges in 
+transformation an unplanned megacity to green city, and finally, the importance of 
+bicycle lane in an unplanned mega city and bicycle lane network design in an unplanned 
+mega city are discussed.  
+3.1 Mobility Problem in an Unplanned Mega City: Dhaka as a Case 
+Study 
+This thesis aim is to enhance transport mobility in an unplanned mega city introducing 
+a bicycle lane. In this section initially, we discuss the history and overview of Dhaka 
+city, then the transportation crisis in Dhaka city and finally, the effect of transportation 
+crisis on other problems. 
+3.1.1 History and Overview of Dhaka City 
+It is mentioned that the concept of a Mega City originated at the end of the 20th century 
+to describe the largest city in the world. Although, literature has little disagreement 
+about the population threshold used as a megacity concept, the UN (2003) defines most 
+precisely: a conurbation of ten million or more inhabitants is a megacity which has now 
+been widely accepted [5].  
+ 
+Since 1971, Dhaka has experienced incredible growth and rapid growth. It is one of the 
+world's only seven cities with a population of over 2.4 percent between 1975 and 2005 
+(UN 2006). In 2011, it was one of the world's top ten mega cities. The developments 
+have unfortunately happened unplanned, especially since the 1990s [5]. The word 
+
+ 
+ 
+11 
+ 
+Dhaka is nowadays mentioned regularly in the most unlivable cities. Dhaka was the 
+world's fastest-growing town between 1950 and 2000 [50]. While population growth 
+has declined recently, it is still the second-largest growth mega city in the world [50]. 
+3.1.2 Transportation Crisis in Dhaka City 
+The mega city has neither efficient public transport nor mass transit [50]. It is probably 
+the world's only mega-city without efficient public transit and public transit [50]. Dhaka 
+has a poorly developed transport system with 200 km of main roads and about 260 km 
+(too few) secondary and collector roads, in addition to 250 km of narrow roads 
+(approximately) [50]. There are many incomplete critical connections in the road 
+network and several regions have insufficient connectivity to the network [50]. Separate 
+bicycling lanes and footpaths are barely available in the city, which enhance the 
+mobility crisis. 
+ 
+There was a time when traffic congestion was only suffered by commuters on the main 
+streets of the city, but now it starts right from the door. Traffic jam has turned into 
+nightmares for daily trips. According to a World Bank report, the average traffic speed 
+in Dhaka has dropped from 21 kilometers per hour (kmph) to 7 kilometers per hour in 
+the last 10 years, and by 2035 the speed could drop to 4 kilometers per hour, which is 
+slower than the walking speed [55]. Another study commissioned by the BRAC 
+Institute of Government and Development indicates that traffic congestion in Dhaka 
+consumes about 5 million working hours a day and costs the country $11.4 billion a 
+year [55]. The financial loss is a measure of the time lost in traffic congestion and the 
+extra hours expended on cars. 
+ 
+It should be noted that there is no adequate and proper routing of our public transport 
+system. In 2016, According to the BRTA, 20,304 new cars were introduced to Dhaka's 
+traffic, which means more than 55 new cars hit the streets every day [55]. As the number 
+of cars increases, there is also an increasing demand for parking space. Unfortunately, 
+however, the parking space in our city is quite inadequate. Many vehicles on the streets 
+are stored. Many buses and trucks are parked on the streets on a regular basis [55].  
+ 
+ 
+
+ 
+ 
+12 
+ 
+According to the Dhaka Metropolitan Police (DMP) Traffic Department, traffic jams 
+have become intolerable in some urban areas over the past few days, including Mirpur-
+12 to Mirpur-10 crossing, Rokeya Sarani, Gulshan, Banani, Badda, Moghbazar, 
+Eskaton, Tejgaon, Airport Road, and Uttara, for a number of reasons, including the 
+ongoing Dhaka International Trade Fair, the building of underground trains and the 
+increase in private transport[56]. Urban analyst and former chairman of UGC Prof 
+Nazrul Islam said traffic jams are gradually deteriorating due to an increase in urban 
+population and the number of small vehicles and lack of effective control measures 
+[56]. “We have built over half dozens of flyovers, but it is not a solution to solve the 
+problem. We will not be able to reduce traffic jams without increasing public transport 
+and ensuring better traffic management", he observed [56]. Transport and urban experts 
+believe that the government should take practical steps to ensure effective mass 
+transportation, restore transportation efficiency, decrease the use of private and small 
+cars, replace micro-buses and mini-busses with single-decker, double-decker, and 
+articulated buses, and extend the city to dramatically alleviate traffic jams without 
+spending huge money [56]. The experts also said that railways and waterways can also 
+be used effectively to relieve road traffic pressure and facilitate trouble-free transport 
+services for the commuters [56].  
+ 
+Figure 3.1 illustrates the traffic jam in the city. Here, three routes are available from 
+Farmgate to the University of Dhaka. During driving mode, it takes around 15 minutes 
+at 06:00 a.m., 18-35 minutes at 10:00 a.m., and 18-40 minutes at 5:00 p.m. on average. 
+On the other side, it takes an average of 45-50 minutes in walking mode. We note that 
+the speed of driving is slightly higher than that of walking. Not only in some areas, but 
+throughout the city, it's the case. 
+ 
+The number of automobiles has been increasing in Dhaka city at the rate of at least 10 
+percent annually, which has been contributing to environmental pollution on the one 
+hand and traffic congestion on the other. This transportation problem enhances other 
+problems like air pollution, noise pollution, fuel consumption, CO2 emission, worst in 
+road conditions, etc. 
+ 
+ 
+ 
+
+ 
+ 
+13 
+ 
+ 
+ 
+ 
+ 
+(A) At 06.00AM 
+(B) At 10.00AM 
+ 
+ 
+(C) At 5.00PM 
+(D) Walking mode 
+Figure 3.1: Farmgate to University of Dhaka routes and time needed in driving mode. (A) At 06.00am. (B) At 
+10.00am. (C) At 5.00pm. (D) Time needed in walking mode. 
+
+回
+of1
+ManikMiaAve
+IndiraRd
+CG
+OCK
+W.Raza
+回
+回
+园
+OFarmgate
+LKISAREA
+Insaf Bal
+Bashundhara City
+&Gen
+spital
+ShoppingComplex
+nthapath
+园
+aHatirheel
+hani Playground
+KALABAGAN
+NEWESK
+12A
+FreeSchool St,
+Kalabegan.1st Ln
+16min
+IANMONDI
+3.9 km
+15min
+OLDE
+5.5 km
+RdNo.8
+回
+17 min
+3.7 km
+Rd8/A
+TA
+RdNo.6
+Off
+回
+H
+HATIRPOOL
+Rd No.5
+ATOLA
+MintoRd
+DhakaCityCollege
+SHAHBAGH
+回
+Kazi Food Industries Ltd.
+KATABON
+Bangladesh
+Dhaka CollegeDhaka
+National Museum
+Shreshtha
+Noor
+hammad
+Dhaka New Market
+UniversityofDhaka
+icCollege
+RAMNA
+et-pilkhana Rd
+回
+Bangla Acad0
+OFarmgate
+IKISAREA
+GA
+KawranBazar
+InsafBarakah
+Bashundhara City
+ShoppingComplex
+&GeneralHos
+Ranthapath@
+Hatirjheel
+上
+ind
+KALABAGAN
+NEWESKATON
+F
+Free SohoolSt
+Kalabagan 1etLn
+DI
+KATHALBAGAN
+OLD ESKATON
+回
+3
+18-35mins
+3.9km
+PARIBAG
+RdNo.6
+HATIRPOOL
+H
+=18-40mins
+Rd No.5
+3.7km
+Dhaka CityCollege
+Baily Rd
+SHAHBAGH
+18-35mins
+Rd
+4.1km
+Bangladesh
+ziFoodIndustriesLtd
+DhakaCollege
+KATABON
+Ant
+HANA
+Dhaka NewMarket
+UniversityofDhakaO
+SaniRd
+RAMNA
+t-Pilkhana Rd
+Google
+BanglaAcademy
+Eden Mohila日
+OFarmgate
+TallabapRd
+IKISAREA
+KawranBazar
+InsafBaral
+Bashundhara City
+ShoppingComplex
+&General
+thap
+atn
+Hatirjheel
+KALABAGAN
+sffofaa
+NEWESKATON
+FroeSchoolSt
+6610-日
+Katabagan1stLn
+H
+KATHALBAGAN
+OLD ESKATON
+3
+20-40mins
+3.9km
+PARIBAGH
+Rd No-6
+HATIRPOOL
+18-40mins
+RdNo.5
+H
+3.7km
+Dhaka City College
+Baily Rd
+SHAHBAGH
+22-40mins
+113419
+4.1km
+回
+Bangladesh
+FoodIndustries Ltd
+DhakaCollege
+KATABON
+11
+ANA
+T
+DhakaNewMarket
+UniversityofDhaka
+SaniRd
+RAMNA
+BanglaAcademy日
+OFarmgate
+TallabieRd
+IKISAREA
+H
+KawranBazar
+Insaf
+Bashundhara City
+ShoppingComplex
+&Ger
+st
+白aHatirjheel
+H
+KALABAGAN
+NEW ESKATON
+Ftee Schodl St
+8010-回
+Kalsbigan1st La
+KATHALBAGAN
+OLDESKATO
+3
+50min
+PARIBAGR
+专T
+3.9km
+RdNo.6
+HATIRPOOL
+Officers'Clut
+H
+RdNo.5
+48min
+ 3.7km
+Shaka CityCollege
+O
+Baily
+49min
+3.8km
+Bangladesh
+IndustriesLtd
+NationalMuseum
+RamnaParl
+DhakaCollege
+KATABON
+DhakaNewMarket
+University of Dhaka0o
+ani
+RAMNA
+PilkhanaRd
+回
+BanglaAcademy 
+ 
+14 
+ 
+3.1.3 Effect of Transportation Crisis on Other Problems 
+Transportation crisis affects the environment badly. It may affect the air pollution, noise 
+pollution, fuel consumption, etc. According to the Department of Environment (DoE), 
+the standard value of the Air Quality Index (AQI) is 50 represents good air quality with 
+little potential to affect public health [51]. But, according to AirVisual information, 
+Dhaka the capital city of Bangladesh has been ranked the worst in the Air Quality Index 
+ 
+ (AQI) valued 309, which is hazardous and would trigger health warnings of emergency 
+conditions [51]. The entire population is more likely to be affected by the enormous 
+number of diseases like nausea, asthma, high blood pressure, heart disease, and cancer. 
+It also impacts the respiratory tract severely and causing irritation. Children's cognitive 
+faculty will be adversely affected by lead exposure, which can also distress the central 
+nervous system, causing hypertension and renal injury. In the last two months, the 
+capitalist has been enjoying just nineteen hours of good air [51]. Diesel-run vehicles 
+account for more than 80 percent of the air pollution in Dhaka as most of them fail to 
+comply with the approved emission standard, said a recently published survey report 
+[52]. 
+ 
+In Dhaka, the average sound level is between 80dB and 110dB in prime areas such as 
+Farmgate, Karwan Bazar, Shahbagh, Gabtoli, and Mohakhali Bus Terminal, says the 
+study report [53]. According to the World Health Organization (WHO), this is almost 
+twice the maximum noise level that can be tolerated by humans – 60dB – without 
+suffering a gradual loss of hearing [53]. According to a recent study conducted by WHO 
+at 45 locations of Dhaka city, most of the traffic points and many of the industrial, 
+residential, commercial, silent and mixed areas are suffering noises exceeding the 
+standard limits of Bangladesh [54]. WHO has also identified several areas as severe 
+red, moderate red, mild red and green zones in terms of noise pollution in Dhaka city 
+[54]. Around 11.7% of the population in Bangladesh have lost their hearing due to noise 
+pollution, says the Development of Environment (DoE) study, which was conducted in 
+2017 [53]. The major sources of noise pollution in urban areas are traffic and loud 
+horns. The DoE found that in Dhaka, 500-1,000 vehicles honk at the same time when 
+stuck in traffic[53]. Around 5% of the world population is facing several kinds of health 
+hazards due to complexities related to noise pollution, According to the WHO [53]. 
+
+ 
+ 
+15 
+ 
+ 
+ 
+There is a scarcity of natural gas and petroleum in Bangladesh also. Gas supplies meet 
+56% of domestic energy demand [57]. Bangladesh has a very limited energy reserve; 
+small amounts of oil, coal and countable natural gas reserves [58]. The country is a net 
+importer of crude oil and petroleum products [57]. 
+3.2 Importance of Bicycle Lane in Mega City 
+In more ways than one, driving a bicycle has a positive impact on the environment. 
+They are also less expensive than other forms of transportation and environment-
+friendly. Bicycles are considered zero-emission vehicles i.e. they do not release any 
+carbon emissions. Bicycles, as vehicles with zero emissions, do not contribute to air 
+pollution. People can have moderate fresh air. They do not contribute to sound 
+pollution. When bicycles are used as a consistent form of travel by a large percentage 
+of the population in a particular area especially in an urban area, there is a great relief 
+on road traffic conditions. Bicycles also have the effect of alleviating parking 
+difficulties in urban areas, because they simply take up so much less space than cars. 
+Low physical activity or Physical inactivity is recognized as one of the country's leading 
+risk factors and the fourth leading cause of deaths due to non-communicable disease 
+(NCDs) worldwide - cardiovascular diseases, chronic lung diseases, heart disease, 
+stroke, diabetes and cancers - and each year contributes to over three million 
+preventable deaths [59]. There may be some physical exercise every day by using bi-
+cycle. Bicycles also offer more freedom of movement without time constraints, 
+crowded and unpleasant conditions and, if desired, the ability to travel alone. So, people 
+can have eco-friendly Travel. Bicycles are lighter and usually cause less damage to the 
+roads than others. This will reduce the number of injuries. So the area would be 
+environment-friendly. Most of the well-organized mega city criteria are met by using 
+bicycles. 
+3.3 Challenges to Increase Mobility in Dhaka City 
+In the case of an unplanned or unorganized city, one of the big issues is that road 
+conditions are not good enough and the cycling lanes & footways are hardly available 
+
+ 
+ 
+16 
+ 
+which is the major cause of worst traffic. On the other hand, noise pollution and traffic 
+congestion are troubling and there is a huge CO2 gas emission occurs.  
+ 
+Between the well-organized city and unplanned city, there is a huge gap in road 
+conditions, footways & cycling lanes, air quality, noise pollution, and traffic 
+congestion. There may have some parameters to increase transport mobility in the city 
+like the construction of separate roads, underground roads, railways, etc. But the 
+construction of those parameters is not a feasible solution because of huge budgets and 
+spaces. There are two alternative low-cost solutions exist, the first one is to make ready 
+the footpaths for routing purpose and the next one is to introduce local lanes with 
+bicycles as vehicles for moving around the city. But the first one is not possible because 
+most of the time, street vendors and hawkers snatch up footways and in some case 
+footways not exist. The next solution is feasible and effective as bicycles have several 
+environmental benefits. 
+ 
+For this purpose, we have to plan a network and always try to use local lanes for routing 
+through one place to another place, if it is not feasible to use local roads for some cases, 
+we will use main roads and will always try to minimize the use of main roads. There 
+exist some constraints to be handled to plan network that we cannot access all possible 
+roads like VIP roads, heavy traffic roads, etc. On the other hand, it is almost impossible 
+to plant more trees to improve air quality and reduce CO2. And most of the noise 
+pollution and traffic congestion is caused by motor vehicle use. 
+3.4 Bicycle Lane Design in an Unplanned Mega City  
+Different approaches have been presented over the past decades to design networks. It 
+is possible to split the solutions into two categories: exact solutions and heuristic 
+solutions. Exact approaches can treat Network Design Problem in a rigorous way which 
+is inefficient when dealing with real-world large-scale networks. And, an approximate 
+yet efficient approach is provided by heuristic approaches, more popular than exact 
+approaches, that have emerged in recent decades which can tackle large-scale real-
+world problems. Without using an exact and heuristic approach, here we present the 
+Physarum-inspired technique which takes into account the constraints to construct the 
+bicycle lane network. Basically, we always try to use local lanes for routing through 
+
+ 
+ 
+17 
+ 
+one place to another place, if it is not feasible to use local roads for some cases, we will 
+use main roads and will always try to minimize the use of main roads.  
+ 
+Compared to previous studies it is noted that the network has only one direction 
+between two nodes, so the stream is only flowing from one node to another, but is never 
+flowing in the opposite direction. However, most roads have the features of double-way 
+traffic in real traffic networks as demonstrated in Fig. 3.2. There is a clear distinction 
+between opposite directions, where flows do not interfere in two directions opposite. 
+Apparently, in the traffic network shown in Fig. 3.2, the initial approach influenced by 
+the Physarum cannot be applied. 
+Here in the following, we discuss the modified Physarum inspired lane design 
+technique. 
+Given a graph 𝐺 = (𝑁, 𝐸), where 
+ 𝑁 denotes a set of 𝑛 cities,  
+ 𝐸 represents a set of 𝑚 connections or linkages. 
+ 
+There is a protoplasmic flow in each link of this model. The two terminals of the link 
+represent two locations of the specified area. One terminal is called the source node, 
+and the other terminal is called the sink node. Protoplasmic flows from the source node 
+into the network and from the sink node out of the network. At each city there is 
+pressure and the amount of flux in each edge is proportional to the difference in pressure 
+between the two terminals of this edge. Specifically, the flux 𝑄𝑖𝑗 in edge (i,j) is given 
+by the modified Hagen-Poiseuille equation below. 
+ 
+Figure 3.2: Real traffic network. 
+ 
+
+ 
+ 
+18 
+ 
+𝑄𝑖𝑗 =  
+𝐷𝑖𝑗
+𝑐𝑖𝑗
+(𝑃𝑖 + 𝑃𝑗)                                                                 (3.1) 
+ 𝐷𝑖𝑗 =  
+𝜋𝑟𝑖𝑗
+4
+8𝜀                                                                         (3.2) 
+In the above equation, 𝐷𝑖𝑗 is the conductivity of the linkage, 𝑐𝑖𝑗/𝐿𝑖𝑗 is the length of the 
+edge,  𝑃𝑖 and 𝑃𝑗 are the pressure of the vertices 𝑖 and 𝑗, 𝑟𝑖𝑗 is the radius of the edge, 
+𝜀 (𝑒𝑝𝑠𝑖𝑙𝑜𝑛) is the coefficient of viscosity. In the case of conductivity(𝐷𝑖𝑗), which is 
+linkage specific, we are using a fixed conductivity value for all the linkages for 
+simplicity. The length(𝑐𝑖𝑗/𝐿𝑖𝑗) is not the direct length from 𝑖 city to 𝑗 city rather we 
+consider all possible path length with no use or hardly use of main roads and we 
+calculate pressure of each city based on the amount of population in that city. As we 
+use initial fixed conductivity value, 𝑟𝑖𝑗 is considered to be the same for all connections.  
+Eq. (3.2) indicates that the tubular thickness( 𝑟𝑖𝑗) of Physarum increases with the 
+conductivity of the tube. Therefore, the conductivity update formula can explain the 
+change in tubular thickness of Physarum as follows,  
+𝑑
+𝑑𝑡 𝐷𝑖𝑗 = 𝑓(|𝑄𝑖����|) −  𝜇𝐷𝑖𝑗 ,                                                   (3.3) 
+here  𝑓(|𝑄𝑖𝑗|) is an increasing function,  𝜇 is a positive constant. In our case, we 
+considered 𝑓(|𝑄𝑖𝑗|) =  |𝑄𝑖𝑗| for simplicity. The equation of conductivity update 
+suggests that conductivity tends to increase with large flux edges. Consequently, the 
+equation of conductivity update reflects the above physiological mechanism. We must 
+first calculate the pressures to calculate the flux and update the edge conductivities. The 
+pressures can be determined using the Poisson equation network below by considering 
+the flux conservation law at each vertex, 
+∑ 𝐷𝑖𝑗
+𝑐𝑖𝑗
+(𝑃𝑖 +  𝑃𝑗) =  {
+    −𝐼0,
+𝑗 = 𝑠𝑜𝑢𝑟𝑐𝑒 
+ +𝐼0,          𝑗 = 𝑠𝑖𝑛𝑘    
+0,            𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
+𝑖 ∈𝑉(𝑗)
+    ,                        (3.4) 
+here  𝑉(𝑗) is the set of vertices linked to vertex j, I0 is the amount of flux flowing into 
+and out of the node of the source.  
+Let the pressure at the sink node be 0, and give an initial value to each edge 
+conductivity, then use Eq. (3.4) to measure the other pressures. After that, we can 
+calculate the amount of flux in each edge using Eq. (3.1), and we can change the 
+conductivity of each edge using Eq. (3.2). According to an edge conductivity threshold 
+value, edges with conductivity lower than this value are cut off from the network. 
+
+ 
+ 
+19 
+ 
+Let’s consider a simple small network consist of only eleven points picked from Dhaka 
+City. Physarum always finds the optimal route network among the eleven nodes and is 
+believed to have achieved a good balance between cost, efficiency, and resilience. Here 
+are the illustrations in the following which are the generalized design using modified 
+Physarum inspired technique. This technique can be applied any real-life traffic 
+network design. In this work, the technique is applied both prominent area centered 
+with Motijheel and entire Dhaka city. 
+In Fig. 3.2, (A) there are 11 node points between Farmgate (node point 2) and the 
+University of Dhaka (node point 10) existing network with both main and local roads. 
+Here, (B) – (E) shows that the network growing process with time (t = iteration) i.e. (B) 
+Network at 10th iteration; (C) Network at 20th iteration; (D) Network at 50th iteration; 
+(E) Final network for 11 nodes. For those cases, we always try to avoid main roads to 
+grow up the networks. And the Fig. 3.3 depicts the possible road network design if all 
+main roads are available. 
+ 
+ 
+Figure 3.3: Sample Region with 11 points. 
+
+日.
+日
+Government
+FAR
+2
+GATE
+Science
+College
+CRd
+IKI'SAREA
+mited
+H
+BashundharaCity
+Insaf BarakahKi
+Rd
+ShoppingComnlex
+&GeneralHos
+apath
+日
+Panthapath
+Hatiriheel
+fosfa
+KAL
+4
+AGAN
+NEW ESKATON
+F
+专T
+KATHAI
+AGAN
+OLD
+ESKAT
+RdNo.8
+LinkRdD
+35
+P
+JAGH
+R
+Rd No.S.
+H
+HAT
+6
+OOL
+MintoRd
+y.College
+Ba
+SHAH
+GH
+oSquare
+NewElephantRd
+日
+Ramna
+KA
+7
+BON
+Bangladesh
+914
+NationalMuseum
+eDhaka
+University
+NewMarket
+ofDhaka
+日
+3
+RA
+10
+NA
+EdenMohila
+College
+Udayan Higher
+Suprem
+SecondarySchool
+ofBang 
+ 
+20 
+ 
+ 
+ 
+ 
+(A) 11 node points (t = 0) 
+ 
+(B) At t = 10 
+Figure 3.4: Physarum inspired network design of 11 nodes. (A) 11 node points. The network is 
+expanding with t. (B) At t = 10. 
+
+11
+-
+6
+7
+1011
+4
+6
+7
+10 
+ 
+21 
+ 
+ 
+ 
+(C) At t = 20 
+ 
+(D) At t = 50 
+Figure 3.4: Physarum inspired network design of 11 nodes. (A) 11 node points. The network is 
+expanding with t. (C) At t = 20. (D) At t = 50.  
+
+6
+7
+104
+L
+7 
+ 
+22 
+ 
+ 
+ 
+ 
+(E) Final network 
+Figure 3.4: Physarum inspired network design of 11 nodes. (E) Final network. 
+
+4
+10 
+ 
+23 
+ 
+ 
+ 
+Figure 3.5: Physarum inspired network design of 11 nodes using main roads (if available). 
+
+4
+6 
+ 
+24 
+ 
+3.5 Significance of Study 
+Modified Physarum Polycephalum Inspired Network Design Technique is used to 
+design any real-life traffic network. In this research, the bicycle lane network is planned 
+using this strategy in a congested mega city Dhaka. This addresses many complicated 
+problems, including the problem of mobility. 
+ 
+Modified Physarum Polycephalum Inspired Network Design Technique holds a 
+significantly different form from the existing Physarum Polycephalum Inspired 
+Network Design Technique. Compared to previous studies it is noted that the network 
+has only one direction between two nodes, so the stream is only flowing from one node 
+to another, but is never flowing in the opposite direction. However, most roads have 
+the features of double-way traffic in real traffic. There is a clear distinction between 
+opposite directions, where flows do not interfere in two directions opposite. Apparently, 
+in real life traffic network, the initial approach influenced by the Physarum cannot be 
+applied. Modified Physarum Polycephalum Inspired Network Design Technique 
+calculates flux and pressure using Eq. (3.1) and Eq. (3.4) respectively as we know that 
+the flow of traffic is not uni-directional rather bi-directional.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+ 
+CHAPTER 4 
+Experimental Studies 
+This chapter experimentally investigates the efficacy of the proposed modified 
+Physarum inspired bicycle lane network design technique. For both a certain portion of 
+Dhaka City and the entire Dhaka City, we are assuming a reduction in the number of 
+buses, cars, taxicabs, and motorcycles. Time saving, fuel saving, user cost saving, and 
+CO2 emission reduction are calculated with some standard average measurement. In 
+this chapter at first, we discuss the prominent points and description of this experiment, 
+then the experimental setting which includes both parameter setting and machine 
+description. In the third section of this chapter, we describe the experimental outcomes 
+and explanation of achievements for both 10km ranged portion and entire Dhaka city 
+are described. 
+4.1 Experimental Settings 
+In the experiment, the number of node points was 29 for both case but 77 linkages/edges 
+are considered for prominent area of Dhaka city and 89 linkages/edges for entire Dhaka 
+city; the length of edges(𝑐𝑖𝑗/𝐿𝑖𝑗) which is not linear and tubular thickness( 𝑟𝑖𝑗) are 
+estimated using the google map; the value of meu(𝜇) was varied from .8 to 1; for certain 
+cases we simply ignore the tubular thickness( 𝑟𝑖𝑗) variations and considered a fixed 
+conductivity value(𝐷𝑖𝑗); it is assumed that, the value of pressure(𝑃𝑖) at each node point 
+is proportional to its population in that area, and a random initial threshold is applied 
+which is increased with the iteration.  
+ 
+The modified Physarum Inspired Bicycle Lane Design was implemented on Visual C++ 
+of Visual Studio 2013. The experiments have been done on a PC (Intel Core i3-5005U 
+CPU @ 2.00 GHz CPU, 2GB NVIDIA GeForce 940M, 4GB RAM) with Windows 10 
+OS.  
+ 
+According to Bangladesh Road Transport Authority (BRTA), on February 04, 2020, 
+there are 127398 registered buses (including microbus, minibus), 293268 registered 
+cars, are 724800 registered motorcycles, and 36600 registered taxicabs currently 
+
+ 
+ 
+25 
+ 
+available in Dhaka city [60]. There may have some unregistered cars & buses and a lot 
+of unregistered motorcycles & taxicabs in the city, we are not considering them. It is 
+assumed that 40 passengers per bus, 1 person per car, taxicab and motorcycle on 
+average. And let’s assume that buses take 20km ride, cars use 10km ride, taxicabs use 
+100km per day and motorcycles ply 15km ride. All those rides are supposed to take 
+place per day. 
+ 
+In this work, both prominent area centered with Motijheel and entire Dhaka city are 
+considered to apply the modified Physarum inspired technique. In this section, the 
+achievements or environmental effects of the prominent area and the entire city of 
+Dhaka with bicycles as vehicle are respectively addressed with the proposed network. 
+4.2 Bicycle Lane Network Design in a Prominent Area 
+At first, a 10km selected prominent area of Dhaka city centered with Motijheel is 
+considered to construct the network using Physarum inspired technique. In this area, 
+we have chosen 29 vital points and numbering those from 1 to 29 arbitrarily. Here, we 
+are considering a 10 km range because the average speed of cycling is 20kmph, so 10 
+km a day can be traveled easily in 30 minutes. And it can also lead to better mental 
+health and energy by bicycling 30 minutes a day [6], [8]. Fig. 4.1 illustrates the selected 
+portion of Dhaka city. Here, (A) depicts the portion of Dhaka city marked within the 
+entire Dhaka city. (B) Shows the 10km range centered with Motijheel. 
+ 
+For 10km ranged area, it is assumed that overall 2% of users switch from car to bicycle 
+and 10% of users switch from motorcycle to bicycle. And a 5% bus and 10% taxicab 
+are being reduced because of using bicycle. Then the atmosphere would change 
+significantly. This section first explains the designed networks and then calculate the 
+time saving and then fuel and cost saving is estimated and finally CO2 emission 
+reduction is calculated. 
+ 
+The locations of prominent area of Dhaka city including traffic pressure is presented in 
+Table 4.1. In this case, we assign some random values as node point's traffic pressure 
+which is proportional to its population. For examples, Taltola, Donia, etc are less and 
+on the other hand Motijheel, Tejgaon, etc are high traffic traffic-pressured area. 
+
+ 
+ 
+26 
+ 
+ 
+ 
+ 
+(A) Bangladesh 
+(B) Dhaka 
+ 
+ 
+(C) Prominent area 
+(D) Node points in Prominent area. 
+1  Motijheel 
+2  Mohakhali 
+3  Gulshan 
+4  Shahinbag 
+5  Tejgaon 
+6  Badda 
+7  EWU 
+8  Bashundhara 
+9  Mogbazar 
+10  Mirbag 
+11  Dhanmondi 
+12  Shahbag 
+13  DU 
+14  Kamlapur  
+15  Ramna 
+16  Kotwali 
+17  Lalbagh 
+18  Khilgaon 
+19  Taltola 
+20  Aftabnagar 
+21  Sadarghat 
+22  Matuail 
+23  Wari 
+24  Golapbag 
+25  Donia 
+26  Rajarbagh 
+27  Sobujbagh 
+28  Green Model Town 
+29  Nandipara 
+Figure 4.1: Selected Dhaka city map. (A) Bangladesh. (B) Dhaka. (C) Prominent area. (D) Node points in Prominent area. 
+ 
+ 
+
+Uzbekistan
+Kyrgyzstan
+Beijing
+北京
+Turkmenistan
+Tajikistan
+China
+Yello
+Afghanistan
+Iran
+Shanghai
+上海
+New.Delhi
+Pakistan
+Ea
+Nepal
+ianGulf
+Bhutan
+Taipei
+台北
+UnitedArab
+Bangladesh
+Emirates
+Taiwan
+India
+My
+imar
+HongKong
+Oman
+Mumbai
+rma)
+香港
+Laos
+Thailand
+South
+Luzon
+Bengaluru
+Vietnam
+China Sea
+23orfedodo
+Bayof Bengal
+Bangkok
+CUMLHIMUICEU
+Arabian Sea
+Cambodia
+Philipp
+AndamanSea
+.HoChi
+Panay
+Gulf of
+MinhCity
+Thailand
+Palawan
+Negro
+SriLanka
+Minda
+Laccadive Sea
+BasilanIsla
+Malaysia
+Kuala Lumpur
+Celebes SHO1
+H07
+N5
+Batshar
+ASSAM
+Kishanganj
+Guwahati
+Nagaon
+Purnia
+AHIDim
+MEGHALAYA
+oShillong
+Sahibganj
+N5
+N2
+Pakur
+Silchar
+Sylhet
+nka
+Raishahi
+N502
+ICGTG
+Hailakandi
+MAI
+T
+N2
+Bangladesh
+N6
+gapur
+N704
+TRIPURAV
+Aizawl
+Dhaka
+N7
+PT
+MIZORAM
+AH
+Jessore
+BENGAL
+Madaripur:
+Satkhira
+Kolkata
+Gk
+N1
+gpur
+Sundarban
+Chittagong
+Forest,
+Bangladesh
+Chandanaish
+eqey
+yangarh
+T5(*
+H
+135153
+N1
+Digha
+Cox'sBazar
+Whaikhyang
+Mrauk-UN302
+UTTARA
+9114
+Hazrat
+N511
+Shahjalal
+N105
+N3
+International
+N501
+Airport
+N301
+MIRPUR
+BASUNDHARA
+Dhaka Zoo
+RESIDENTIAL
+ 
+N3
+AREA
+Dhaka
+Jalshiri.Abason
+raid
+GULSHAN
+N5
+Gabtoli
+Z5069
+5114
+MOHAKHALI
+BADDA
+R202
+MOHAMMADPUR
+TEJGAON
+KHILGAON
+ROTST
+Boshila
+DHANMONDI
+R
+ty
+RAMNA
+MOTIJHEEL
+Chanpara
+尼
+b
+N2
+LalbaghFort
+Hizla
+R110
+KOTWALI
+N2
+Matuail
+Keraniganji
+8N
+R820
+R820
+N1
+Shiddhirganj
+Z5069
+R810
+R111
+Ruhitpur
+Bashundhara
+Kadamtoli
+Riverview
+Baghair
+Ekuria
+R810BARIDHARA
+MadaniAve
+Banani
+Suvastu Nazar Valley
+ShoppingComplex
+SHER-E-BANGLA
+NAGAR
+National
+Parliament House
+LALMATIA
+H
+DhakaNewMarket
+2
+Madina Filling Station
+R820
+R110
+Buriganga River
+R820
+N1
+Z1102
+Keraniganj
+21
+NB
+R820
+RanaCNG&
+GASStation 
+ 
+27 
+ 
+4.2.1 Network Design 
+Here in the planned network, the distance is not the linear distance between two node 
+points rather distance is calculated using google map. And the time is calculated 
+considering standard speed 20kmph for bicycle in minutes. 
+ 
+In Fig. 4.2, (A) there are 29 node points around 10km range centered with Motijheel 
+(node point 1) existing network with both main and local roads. Figure (B) – (E) shows 
+that the network growing process with time (t = iteration) i.e. (B) Network in 10th 
+iteration; (C) Network 20th iteration; (D) Network 50th iteration; (E) Finale network. 
+For those cases, we always try to avoid main roads to grow up the networks. In Fig. 4.3 
+Network is planned with main roads (if available). 
+ 
+Table 4.1: The locations of prominent area of Dhaka city including traffic pressure. 
+Sl 
+Location 
+Traffic Pressure 
+01 
+Motijheel 
+9 
+02 
+Mohakhali 
+8 
+03 
+Gulshan 
+5 
+04 
+Shahinbag 
+8 
+05 
+Tejgaon 
+9 
+06 
+Badda 
+4 
+07 
+EWU 
+5 
+08 
+Bashundhara 
+5 
+09 
+Mogbazar 
+5 
+10 
+Mirbag 
+5 
+11 
+Dhanmondi 
+7 
+12 
+Shahbag 
+9 
+13 
+DU 
+9 
+14 
+Kamlapur 
+8 
+15 
+Ramna 
+5 
+16 
+Kotwali 
+3 
+17 
+Lalbagh 
+4 
+18 
+Khilgaon 
+6 
+19 
+Taltola 
+3 
+20 
+Aftabnagar 
+4 
+21 
+Sadarghat 
+5 
+22 
+Matuail 
+8 
+23 
+Wari 
+7 
+24 
+Golapbag 
+5 
+25 
+Donia 
+3 
+26 
+Rajarbagh 
+5 
+27 
+Sobujbagh 
+5 
+28 
+Green Model Town 
+4 
+29 
+Nandipara 
+6 
+ 
+ 
+ 
+
+ 
+ 
+28 
+ 
+ 
+ 
+(A) 10km range existing road network 
+ 
+(B) When t=10 
+ 
+Figure 4.2: Network design using modified Physarum inspired technique. (A) 10km range existing road 
+network. (B) When t=10. 
+
+12
+26
+16
+2127
+16 
+ 
+29 
+ 
+ 
+ 
+ 
+(C) When t=20 
+ 
+(D) When t=50 
+ 
+Figure 4.2: Network design using modified Physarum inspired technique. (C) When t=20. (D) When t=50. 
+
+27
+16
+219
+12
+16 
+ 
+30 
+ 
+ 
+ 
+ 
+(E) Final Network 
+Figure 4.2: Final network design using modified Physarum inspired technique.  
+
+27
+26 
+ 
+31 
+ 
+ 
+ 
+Figure 4.3: Network design using modified Physarum inspired technique using main roads (if available). 
+
+27
+2
+16 
+ 
+32 
+ 
+Here the Table 4.2 shows routes of all node points from starting node point 1. For 
+example, the path 1-9-10-2-3 means that we have to cross node points 9, 10 and 2 in 
+order to go into destination node point 3 from starting node point 1. The minimum 
+distance is considered for accessing any node here. For example, the destination node 
+point 3 can be accessed using the routes 1-9-10-2-3, 1-20-7-6-2-3, 1-12-5-4-3 and so 
+on but the minimum distanced one is considered for calculation. 
+4.2.2 Effectiveness Analysis 
+Driving a paddled-bicycle has a more than one beneficial environmental effect. For 
+many cases bicycles take less time to ride, no fuel usage, saving user money, CO 2 
+emission reduction in unplanned mega city and do not contribute to air pollution, sound 
+pollution, great relief on road traffic conditions, alleviating parking difficulties in urban 
+areas, cause less damage to the roads, get relief of some non-communicable disease 
+Table 4.2: Routes from node point 1. 
+Des. Node 
+Point 
+Routes 
+Distance 
+(km) 
+Estimated Travel 
+Time (min) 
+2 
+1-9-10-2 
+8.55 
+25.65 
+3 
+1-9-10-2-3 
+11.25 
+33.75 
+4 
+1-9-4 
+7.3 
+21.9 
+5 
+1-5 
+7.5 
+22.5 
+6 
+1-9-10-7-6 
+8.2 
+24.6 
+7 
+1-9-10-7 
+6.9 
+20.7 
+8 
+1-8 
+6.85 
+20.55 
+9 
+1-9 
+2.9 
+8.7 
+10 
+1-9-10 
+4.4 
+13.2 
+11 
+1-12-13-11 
+9.6 
+28.8 
+12 
+1-12 
+5.3 
+15.9 
+13 
+1-12-13 
+6.4 
+19.2 
+14 
+1-14 
+5.7 
+17.1 
+15 
+1-15 
+2.4 
+7.2 
+16 
+1-23-16 
+4.5 
+13.5 
+17 
+1-23-16-17 
+8.1 
+24.3 
+18 
+1-20-18 
+9 
+27 
+19 
+1-19 
+3.7 
+11.1 
+20 
+1-20 
+6.2 
+18.6 
+21 
+1-23-21 
+4.5 
+13.5 
+22 
+1-23-21-25-22 
+10.3 
+30.9 
+23 
+1-23 
+2.8 
+8.4 
+24 
+1-14-24 
+7.15 
+21.45 
+25 
+1-23-21-25 
+7.1 
+21.3 
+26 
+1-14-26 
+6.7 
+20.1 
+27 
+1-14-26-27 
+9.9 
+29.7 
+28 
+1-14-28 
+7.9 
+23.7 
+29 
+1-14-26-29 
+8.1 
+24.3 
+ 
+ 
+ 
+
+ 
+ 
+33 
+ 
+(NCDs) for have some physical exercise every day. The considering factors of this 
+paper are only time saving, fuel and user money saving and CO2 emission reduction.  
+4.2.2.1 Time Saving 
+As we mentioned that, according to a World Bank report, the average traffic speed in 
+Dhaka has dropped from 21 kilometers per hour (kmph) to 7 kilometers per hour in the 
+last 10 years, and by 2035 the speed could drop to 4 kilometers per hour, which is 
+slower than the walking speed. In our constructed network, we mainly try to avoid main 
+roads or minimum usage of main roads if requires. So, here traffic jam is hardly 
+available. For convenience, we assume no traffic jam exists in the following calculation 
+of time.  
+ 
+In Table 4.3, the time needed for both cars and buses are listed in three different times 
+(at 6:00 AM, 10:00 AM, and 4:00 PM) calculated from google map and the time needed 
+for a bicycle is always constant. Here, we measure the distance and time of all the node 
+point from center node point 1. 
+ 
+The time required for a car, taxicab and motorcycle are almost the same that’s why we 
+don’t mention it differently in the Table 4.3. Here we notice that at the morning 6:00 
+AM the travel time is less because of minimal traffic in the roads, at 10:00 AM (peak 
+hour) when traffic jam occurs severe the need time to travel is huge and at 4:00 PM 
+there exist traffic jam also but sometimes a bit less than peak hour. This condition is 
+true for all types of cars, buses, taxicabs, etc. For the prominent area, the gross working 
+hours saving are described in the following Table 4.4. So, around 152216 working 
+hours per day can be saved using bicycle in the prominent area. 
+4.2.2.2 Fuel and Cost Saving 
+The cost of installation or repair is not listed here rather we considering only running 
+cost. Because whenever we switch from car to bicycle there would be a significant 
+reduction in costs. 
+ 
+Here the mileage of car and taxicab is assumed at 20kmpl and 25kmpl respectively with 
+diesel and 65tk per liter diesel. On the other hand, cycling has no cost. The bus mileage  
+considers 5kmpl with diesel and motorcycle has 50kmpl with petrol. 
+
+ 
+ 
+34 
+ 
+ 
+In Table 4.4, the needed fuel for cars, taxicabs, motorcycles, and buses is calculated 
+and there is no running cost & fuel cost for the bicycle. In case of bus, the per km fare 
+is fixed by BRTA of 1.7tk and we assumed that taxicab fare is 50tk per km. Here, the 
+distance of all the node points are measured from center node point 1. 
+ 
+Table 4.3: Time comparison in car, bus and bicycle considering from node point 1. 
+Node 
+Point 
+Dista
+nce 
+(km) 
+Car & Taxicab & Motor cycle 
+Bus 
+Bicycle  
+6:00am 
+(min) 
+10:00am 
+(min) 
+4:00pm 
+(min) 
+6:00am 
+(min) 
+10:00am 
+(min) 
+4:00pm 
+(min) 
+Dista
+nce 
+(km) 
+Time 
+(min) 
+2 
+8.6 
+18 
+39 
+36 
+20 
+43 
+40 
+8.55 
+25.65 
+3 
+8.9 
+19 
+42 
+39 
+21 
+46 
+43 
+11.25 
+33.75 
+4 
+7.1 
+14 
+33 
+30 
+16 
+36 
+33 
+7.3 
+21.9 
+5 
+5.1 
+12 
+27 
+26 
+14 
+30 
+29 
+7.5 
+22.5 
+6 
+7.6 
+15 
+36 
+36 
+17 
+39 
+39 
+8.2 
+24.6 
+7 
+7.6 
+15 
+36 
+36 
+17 
+39 
+39 
+6.9 
+20.7 
+8 
+4.8 
+11 
+24 
+24 
+13 
+27 
+27 
+6.85 
+20.55 
+9 
+3.6 
+9 
+18 
+15 
+11 
+20 
+17 
+2.9 
+8.7 
+10 
+4 
+10 
+20 
+18 
+12 
+22 
+20 
+4.4 
+13.2 
+11 
+6.1 
+12 
+30 
+30 
+14 
+33 
+33 
+9.6 
+28.8 
+12 
+3.4 
+9 
+23 
+23 
+11 
+26 
+26 
+5.3 
+15.9 
+13 
+3.8 
+10 
+24 
+21 
+12 
+27 
+24 
+6.4 
+19.2 
+14 
+1.2 
+3 
+11 
+11 
+5 
+12 
+12 
+5.7 
+17.1 
+15 
+3.1 
+8 
+18 
+18 
+10 
+20 
+20 
+2.4 
+7.2 
+16 
+3.2 
+8 
+27 
+33 
+10 
+29 
+35 
+4.5 
+13.5 
+17 
+6.1 
+13 
+33 
+30 
+15 
+36 
+33 
+8.1 
+24.3 
+18 
+7.2 
+14 
+53 
+48 
+16 
+56 
+51 
+9 
+27 
+19 
+3 
+8 
+21 
+18 
+10 
+23 
+20 
+3.7 
+11.1 
+20 
+10 
+24 
+42 
+38 
+26 
+47 
+43 
+6.2 
+18.6 
+21 
+3.5 
+9 
+36 
+36 
+11 
+38 
+38 
+4.5 
+13.5 
+22 
+8.3 
+18 
+33 
+30 
+20 
+37 
+34 
+10.3 
+30.9 
+23 
+2.5 
+6 
+21 
+21 
+8 
+23 
+23 
+2.8 
+8.4 
+24 
+4.2 
+10 
+27 
+27 
+12 
+30 
+30 
+7.15 
+21.45 
+25 
+5.7 
+13 
+29 
+27 
+15 
+32 
+30 
+7.1 
+21.3 
+26 
+4.7 
+11 
+29 
+29 
+13 
+32 
+32 
+6.7 
+20.1 
+27 
+3.1 
+8 
+15 
+12 
+10 
+17 
+14 
+9.9 
+29.7 
+28 
+5.8 
+12 
+42 
+39 
+14 
+45 
+42 
+7.9 
+23.7 
+29 
+5.4 
+12 
+33 
+33 
+14 
+36 
+36 
+8.1 
+24.3 
+ 
+ 
+ 
+Table 4.4: Time saving per day. 
+Transit 
+% of Transit 
+Reduction 
+Transit 
+Reduces 
+Riding Distance 
+(km) 
+Time saves 
+(min) 
+Bus 
+5% 
+6370 
+127400 
+709800 
+Car 
+2% 
+5865 
+58650 
+326764 
+Taxicab 
+10% 
+3660 
+366000 
+2039143 
+Motor cycle 
+10% 
+72480 
+1087200 
+6057257 
+Total time saving  
+9132964 
+ 
+ 
+
+ 
+ 
+35 
+ 
+The gross fuel saving and user cost saving for motijheel area are calculated for buses, 
+cars, taxicabs, and motorcycles and described in the Table 4.6. So, around 64797 liters 
+fuel and 20.6 million user money costs can be saved per day using bicycle in the 
+motijheel area. 
+  
+ 
+Table 4.5: Cost comparison in car, bus and bicycle considering from node point 1. 
+Node 
+Point 
+Dista
+nce 
+(km) 
+Car 
+Taxicabs 
+Motor cycle 
+Bus 
+Bicycle  
+Fuel 
+(litre) 
+User 
+Cost 
+(tk) 
+Fuel 
+(litre) 
+User 
+Cost 
+(tk) 
+Fuel 
+(litre) 
+User 
+Cost 
+(tk) 
+Fuel 
+(litre) 
+User 
+Cost 
+(tk) 
+Dista
+nce 
+(km) 
+Cost 
+(tk) 
+2 
+8.6 
+0.43 
+27.95 
+0.34 
+430 
+0.17 
+15.31 
+ 
+ 
+ 
+ 
+As we 
+assume 
+bicycles 
+reduces 
+5% of 
+the total 
+bus in 
+Dhaka 
+city. 
+ 
+The fuel 
+consump
+tion is 
+reduced  
+=  
+6370×20
+×1/5 
+litres 
+ 
+= 25480 
+litres 
+14.62 
+8.55 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+N/A 
+3 
+8.9 
+0.45 
+28.93 
+0.36 
+445 
+0.18 
+15.84 
+15.13 
+11.25 
+4 
+7.1 
+0.36 
+23.08 
+0.28 
+355 
+0.14 
+12.64 
+12.07 
+7.3 
+5 
+5.1 
+0.26 
+16.58 
+0.2 
+255 
+0.1 
+9.08 
+8.67 
+7.5 
+6 
+6.8 
+0.34 
+22.1 
+0.27 
+340 
+0.14 
+12.1 
+11.56 
+8.2 
+7 
+7.6 
+0.38 
+24.7 
+0.3 
+380 
+0.15 
+13.53 
+12.92 
+6.9 
+8 
+4.8 
+0.24 
+15.6 
+0.19 
+240 
+0.1 
+8.54 
+8.16 
+6.85 
+9 
+3.6 
+0.18 
+11.7 
+0.14 
+180 
+0.07 
+6.41 
+6.12 
+2.9 
+10 
+4 
+0.2 
+13 
+0.16 
+200 
+0.08 
+7.12 
+6.8 
+4.4 
+11 
+6.1 
+0.31 
+19.83 
+0.24 
+305 
+0.12 
+10.86 
+10.37 
+9.6 
+12 
+3.4 
+0.17 
+11.05 
+0.14 
+170 
+0.07 
+6.05 
+5.78 
+5.3 
+13 
+3.8 
+0.19 
+12.35 
+0.15 
+190 
+0.08 
+6.76 
+6.46 
+6.4 
+14 
+1.2 
+0.06 
+3.9 
+0.05 
+60 
+0.02 
+2.14 
+2.04 
+5.7 
+15 
+3.1 
+0.16 
+10.08 
+0.12 
+155 
+0.06 
+5.52 
+5.27 
+2.4 
+16 
+3.2 
+0.16 
+10.4 
+0.13 
+160 
+0.06 
+5.7 
+5.44 
+4.5 
+17 
+6.1 
+0.31 
+19.83 
+0.24 
+305 
+0.12 
+10.86 
+10.37 
+8.1 
+18 
+7.2 
+0.36 
+23.4 
+0.29 
+360 
+0.14 
+12.82 
+12.24 
+9 
+19 
+3 
+0.15 
+9.75 
+0.12 
+150 
+0.06 
+5.34 
+5.1 
+3.7 
+20 
+10 
+0.5 
+32.5 
+0.4 
+500 
+0.2 
+17.8 
+17 
+6.2 
+21 
+3.5 
+0.18 
+11.38 
+0.14 
+175 
+0.07 
+6.23 
+5.95 
+4.5 
+22 
+8.3 
+0.42 
+26.98 
+0.33 
+415 
+0.17 
+14.77 
+14.11 
+10.3 
+23 
+2.5 
+0.13 
+8.13 
+0.1 
+125 
+0.05 
+4.45 
+4.25 
+2.8 
+24 
+4.2 
+0.21 
+13.65 
+0.17 
+210 
+0.08 
+7.48 
+7.14 
+7.15 
+25 
+5.7 
+0.29 
+18.53 
+0.23 
+285 
+0.11 
+10.15 
+9.69 
+7.1 
+26 
+4.7 
+0.24 
+15.28 
+0.19 
+235 
+0.09 
+8.37 
+7.99 
+6.7 
+27 
+3.1 
+0.16 
+10.08 
+0.12 
+155 
+0.06 
+5.52 
+5.27 
+9.9 
+28 
+5.8 
+0.29 
+18.85 
+0.23 
+290 
+0.12 
+10.32 
+9.86 
+7.9 
+29 
+5.4 
+0.27 
+17.55 
+0.22 
+270 
+0.11 
+9.61 
+9.18 
+8.1 
+ 
+ 
+ 
+Table 4.6: Cost saving per day. 
+Transit 
+% of Transit 
+Reduction 
+Transit 
+Reduces 
+Riding 
+Distance (km) 
+Fuel 
+(litres) 
+User Cost 
+(tk) 
+Bus 
+5% 
+6370 
+127400 
+25480 
+216580 
+Car 
+2% 
+5865 
+58650 
+2933 
+190613 
+Taxicab 
+10% 
+3660 
+366000 
+14640 
+18300000 
+Motor cycle 
+10% 
+72480 
+1087200 
+21744 
+1935216 
+Total cost saving  
+64797 
+20642409 
+ 
+ 
+
+ 
+ 
+36 
+ 
+4.2.2.3 CO2 Emission Reduction 
+In the calculation, 887 g/km, 258 g/km, 237 g/km and 40 g/km are the considered 
+amount of CO2 emission in 1km ride of bus, car, taxicab and motorcycle respectively. 
+For motijheel, the gross CO2 emission reduction is described in the following Table 4.7. 
+In total, around 6.5 × 105 kg CO2 emission is reduced in the motijheel area per day.   
+4.3 Bicycle Lane Network Design for Entire Dhaka City 
+Here, 29 important locations of entire Dhaka city are selected and numbering those 
+from 1 to 29 randomly, which are depicted in following Fig. 4.4. As it is considered 
+that the paddled-bicycle for routing 10km ranged area, it is not feasible to move through 
+all over the Dhaka city using paddled-bicycle. But in case of electric bicycle, it is 
+possible. That’s why electric bicycles are considered as vehicles for entire Dhaka city. 
+ 
+The general data of Dhaka city including location, population and traffic pressure are 
+described in the Table 4.8. In this case, the consideration is that the node point's traffic 
+pressure is proportional to its population. 
+ 
+For entire Dhaka city, it is assumed that 5% of users switch from car to electrical 
+motorcycle or bicycle and 50% of users switch from motorcycle to electrical 
+motorcycle or bicycle. And a 10% bus and 20% taxicab are being reduced because of 
+using electric motorcycles or bicycles. Then there will be some noteworthy change in 
+the environment. 
+4.3.1 Network Design 
+The constructed network of selected 29 points of Dhaka city is illustrated in Fig. 4.5. 
+In the Table 4.9, routes are shown to all node points from starting node point 7. For  
+Table 4.7: CO2 emission reduction per day. 
+Transit 
+% of Transit 
+Reduction 
+Transit 
+Reduces 
+Riding 
+Distance (km) 
+CO2 Emission 
+(g) 
+Bus 
+5% 
+6370 
+127400 
+1.13 × 108 
+Car 
+2% 
+5865 
+58650 
+1.51 × 107 
+Taxicab 
+10% 
+3660 
+366000 
+8.67 × 107 
+Motor cycle 
+10% 
+72480 
+1087200 
+4.35 × 107 
+Total CO2 saving  
+2.58 × 108 
+ 
+ 
+
+ 
+ 
+37 
+ 
+ 
+ 
+ 
+01 Uttara 
+02 Mirpur 
+03 Uttara ABM City 
+04 Basundhara R/A 
+05 Khilkhet 
+06 Cantonment 
+07 Gulshan 
+08 Badda 
+09 Mohakhali 
+10 Tejgaon 
+11 Motijheel 
+12 Khilgoan 
+13 Gabtoli 
+14 Mohammadpur 
+15 Dhanmondi 
+16 Shahbagh 
+17 Matuail 
+18 Kotwali 
+19 New Market 
+20 Baridhara 
+21 Banani 
+22 Monipur 
+23 Sher-E-Bangla Nagar 
+24 Airport 
+25 Poradia 
+26 Madarbari 
+27 Beraid 
+28 NutanPara 
+29 Pagla Rail station 
+ 
+Figure 4.3: Entire Dhaka city. 
+ 
+ 
+
+Asnua
+N302
+N501
+R303
+N511
+N302
+N501
+Dhaka Zoo
+JalshiriAb
+23
+R20
+OL
+Boshila
+1.5
+28
+Chanpa
+LalbaghFort
+b
+Hizla
+R110
+Keraniganj
+R820
+N8
+820
+N1
+Shiddhirganj
+5069
+RB10
+Bashundhara
+R111
+pur
+Kada
+Riverview
+Baghair
+Ekuria
+N8
+folafe
+29
+eshviariRN
+Kalakandi
+Gode
+Zazira 
+ 
+38 
+ 
+ 
+example, the route 7-6-24-5-1 means that we have to pass node points 6, 26 and 5 in 
+order to arrive destination node point 1 from starting node point 7. We are considering 
+the minimum distance for accessing any node here. For example, the destination node 
+point 1 can be accessed using the routes 7-6-24-5-1, 7-6-4-26-5-1, 7-21-22-2-24-5-1 
+and so on. But the minimum distanced route is considered for calculation. 
+4.3.2 Effectiveness Analysis 
+Using an electric-bicycle has a more than one beneficial environmental effect. For most 
+of the cases bicycles take less time to ride, no fuel usage, saving user money, CO2 
+emission reduction in unplanned mega city and very less contribution to air pollution, 
+sound pollution, great relief on road traffic conditions, alleviating parking difficulties 
+in urban areas, cause less damage to the roads, get relief of some non-communicable 
+disease (NCDs) for have some physical exercise every day. The considering factors of  
+Table 4.8: The general data on locations in Dhaka city, including population, area, and traffic pressure. 
+Sl 
+Location 
+Population 
+Area (km²) 
+Traffic Pressure 
+01 
+Uttara 
+345,097 
+36.91 
+10 
+02 
+Mirpur 
+274,530 
+4.71 
+8 
+03 
+Uttara ABM City 
+145,097 
+27.91 
+5 
+04 
+Bashundhara R/A 
+274,200 
+13.54 
+8 
+05 
+Khilkhet 
+130,053 
+15.88 
+5 
+06 
+Cantonment 
+117,464 
+14.47 
+4 
+07 
+Gulshan 
+145,969 
+ 8.85 
+5 
+08 
+Badda 
+157,924 
+16.78 
+5 
+09 
+Mohakhali 
+145,969 
+8.85 
+5 
+10 
+Tejgaon 
+148,255 
+2.46 
+5 
+11 
+Motijheel 
+225,999 
+3.69 
+7 
+12 
+Khilgaon 
+327,717 
+13.8 
+9 
+13 
+Gabtoli 
+198,723 
+4.98 
+6 
+14 
+Mohammadpur 
+355,843 
+11.65 
+10 
+15 
+Dhanmondi 
+147,643 
+2.86 
+5 
+16 
+Shahbag 
+74,113 
+3.49 
+3 
+17 
+Matuail 
+125,312 
+19.36 
+4 
+18 
+Kotwali 
+210,504 
+0.67 
+6 
+19 
+New Market 
+66,439 
+1.67 
+2 
+20 
+Baridhara 
+105,969 
+5.45 
+4 
+21 
+Banani 
+145,969 
+8.85 
+5 
+22 
+Monipur 
+274,530 
+4.71 
+8 
+23 
+Sher-E-Bangla Nagar 
+248,871 
+5.25 
+7 
+24 
+Airport 
+130,053 
+15.88 
+5 
+25 
+Poradia 
+52,014 
+20.09 
+2 
+26 
+Madarbari 
+93,153 
+12.65 
+3 
+27 
+Beraid 
+157,924 
+16.78 
+5 
+28 
+NutanPara 
+125,312 
+19.36 
+4 
+29 
+Pagla Rail station 
+194,019 
+246.21 
+6 
+ 
+ 
+ 
+
+ 
+ 
+39 
+ 
+this paper is only time saving, fuel and user money saving and CO2 emission reduction. 
+4.3.2.1 Time Saving 
+In the following measurements, 7 kilometers per hour (kmph) for the buses, cars, 
+taxicabs & motorcycles which is the average speed of traffic in Dhaka city and 40 km/h 
+for electric bicycles using our constructed network. We are taken into consideration our 
+constructed network as jam-free.  In the Table 4.10, the time necessary for cars and 
+buses are listed in three different times (at 6:00 AM, 10:00 AM, and 4:00 PM) 
+calculated from google map and the time needed for a bicycle is always constant. The 
+required time for taxicabs and motorcycles is considered to be the same as the time 
+needed for a car. Here, we measure the distance and time of all the node point from 
+center node point 7. 
+ 
+Table 4.9: Routes from node point 7. 
+Des. Node 
+Point 
+Routes  
+Distance 
+(km) 
+Estimated Travel 
+Time(min) 
+1 
+7-6-24-5-1 
+15.65 
+23.48 
+2 
+7-21-22-2 
+9.89 
+14.84 
+3 
+7-6-24-5-26-3 
+19.51 
+29.27 
+4 
+7-6-4 
+9.02 
+13.53 
+5 
+7-6-24-5 
+11.9 
+17.85 
+6 
+7-6 
+3.95 
+5.93 
+8 
+7-20-8 
+3.17 
+4.76 
+9 
+7-9 
+2.17 
+3.26 
+10 
+7-9-10 
+5.98 
+8.97 
+11 
+7-9-10-16-11 
+10.61 
+15.92 
+12 
+7-20-12 
+9.04 
+13.56 
+13 
+7-21-13 
+9.44 
+14.16 
+14 
+7-23-14 
+7.31 
+10.97 
+15 
+7-23-14-15 
+11.55 
+17.33 
+16 
+7-9-10-16 
+8.7 
+13.05 
+17 
+7-9-28-17 
+15.61 
+23.42 
+18 
+1-20-18 
+14.78 
+22.17 
+19 
+7-9-10-19 
+9.87 
+14.81 
+20 
+7-20 
+2.12 
+3.18 
+21 
+7-21 
+1.76 
+2.64 
+22 
+7-21-22 
+7.34 
+11.01 
+23 
+7-23 
+4.07 
+6.11 
+24 
+7-6-24 
+8.55 
+12.83 
+25 
+7-6-4-25 
+18.72 
+28.08 
+26 
+7-6-24-5-26 
+15.5 
+23.25 
+27 
+7-20-27 
+11.2 
+16.8 
+28 
+7-9-28 
+9.41 
+14.12 
+29 
+7-9-28-17-29 
+22.02 
+33.03 
+ 
+ 
+ 
+
+ 
+ 
+40 
+ 
+ 
+ 
+Figure 4.4: Constructed network 29 points of Dhaka city. 
+ 
+ 
+
+25 
+ 
+41 
+ 
+In the Table 4.10, we have noticed that at the morning 6:00 AM the travel time is less 
+because of minimal traffic in the roads, at 10:00 AM (peak hour) when traffic jam 
+occurs severe amount the need time to travel is huge and at 4:00 PM there exist traffic 
+jam also but sometimes a bit less than peak hour or sometimes a bit high. This condition 
+is true for all types of cars, buses, taxicabs, etc. For the entire Dhaka city, the gross 
+working hours saving are described in the following Table 4.11. So, around .2 million 
+working hours per day using bicycle in the motijheel area. 
+Table 4.10: Time comparison in car, bus and bicycle considering from node point 7. 
+Node 
+Point 
+Dista
+nce 
+(km) 
+Car & Taxicab & Motor cycle 
+Bus 
+Bicycle  
+6:00am 
+(min) 
+10:00am 
+(min) 
+4:00pm 
+(min) 
+6:00am 
+(min) 
+10:00am 
+(min) 
+4:00pm 
+(min) 
+Dista
+nce 
+(km) 
+Time 
+(min) 
+1 
+14 
+24 
+42 
+45 
+26 
+50 
+51 
+15.65 
+23.48 
+2 
+7.5 
+16 
+33 
+36 
+18 
+43 
+42 
+9.89 
+14.84 
+3 
+18.2 
+40 
+75 
+75 
+42 
+82 
+81 
+19.51 
+29.27 
+4 
+10.2 
+22 
+42 
+36 
+22 
+42 
+42 
+9.02 
+13.53 
+5 
+8.7 
+12 
+24 
+24 
+14 
+31 
+30 
+11.9 
+17.85 
+6 
+6 
+10 
+26 
+24 
+12 
+32 
+30 
+3.95 
+5.93 
+8 
+4.7 
+8 
+30 
+27 
+9 
+35 
+32 
+3.17 
+4.76 
+9 
+3.5 
+7 
+24 
+22 
+9 
+28 
+24 
+2.17 
+3.26 
+10 
+5.5 
+10 
+25 
+24 
+12 
+31 
+28 
+5.98 
+8.97 
+11 
+9.1 
+18 
+45 
+42 
+20 
+52 
+48 
+10.61 
+15.92 
+12 
+12 
+40 
+60 
+57 
+42 
+68 
+65 
+9.04 
+13.56 
+13 
+9.7 
+22 
+44 
+45 
+24 
+51 
+51 
+9.44 
+14.16 
+14 
+8.3 
+14 
+36 
+33 
+16 
+44 
+39 
+7.31 
+10.97 
+15 
+8.7 
+16 
+33 
+36 
+17 
+41 
+42 
+11.55 
+17.33 
+16 
+8 
+16 
+39 
+39 
+18 
+47 
+45 
+8.7 
+13.05 
+17 
+16.6 
+30 
+63 
+60 
+32 
+71 
+66 
+15.61 
+23.42 
+18 
+11.4 
+24 
+62 
+60 
+26 
+60 
+66 
+14.78 
+22.17 
+19 
+8.7 
+18 
+42 
+42 
+20 
+50 
+48 
+9.87 
+14.81 
+20 
+2.2 
+5 
+12 
+11 
+6 
+15 
+14 
+2.12 
+3.18 
+21 
+2.3 
+5 
+17 
+15 
+7 
+17 
+15 
+1.76 
+2.64 
+22 
+6.8 
+20 
+36 
+33 
+22 
+44 
+39 
+7.34 
+11.01 
+23 
+6.6 
+12 
+24 
+23 
+14 
+32 
+29 
+4.07 
+6.11 
+24 
+8.7 
+12 
+24 
+24 
+14 
+32 
+30 
+8.55 
+12.83 
+25 
+15.9 
+30 
+68 
+71 
+32 
+75 
+77 
+18.72 
+28.08 
+26 
+14.7 
+30 
+63 
+60 
+32 
+71 
+66 
+15.5 
+23.25 
+27 
+8.5 
+14 
+36 
+33 
+16 
+45 
+39 
+11.2 
+16.8 
+28 
+10.2 
+24 
+63 
+60 
+26 
+71 
+66 
+9.41 
+14.12 
+29 
+19.1 
+35 
+79.5 
+75 
+37 
+87 
+81 
+22.02 
+33.03 
+ 
+ 
+ 
+Table 4.11: Time saving per day. 
+Transit 
+% of Transit 
+Reduction 
+Transit 
+Reduces 
+Riding 
+Distance (km) 
+Time Saves 
+(min) 
+Bus 
+10% 
+12740 
+254796 
+7.2 × 107 
+ 
+ 
+ 
+ 
+ 
+Car 
+5% 
+14663 
+146630 
+1.04 × 106 
+Taxicab 
+20% 
+7320 
+732000 
+5.2 × 106 
+Motor cycle 
+50% 
+362400 
+5436000 
+3.8 × 107 
+Total time saving  
+1.16 × 108 
+ 
+ 
+
+ 
+ 
+42 
+ 
+Time Saving Explanation: 
+Total Bus ride reduce: 0. 1 ×  127398 ×  20 km = 254796 km 
+Time saving for Bus: 254796 × 40 × ( 8.57 – 1.5) mins = 7.2 × 107 mins  
+= 1.2 × 106 hours = 50000 day  
+Total Car ride reduce: 0.05 × 293268 × 10 km = 146634 km 
+Time saving for Car: 146634 × (8.57 – 1.5) mins = 1.04 × 106 mins 
+ 
+ 
+ 
+= 1.73 × 104 hours = 720 days 
+Total Taxicab ride reduce: 0.2 × 36600 × 100 km = 732000 km 
+Time saving for Taxicab: 732000 × (8.57 – 1.5) mins = 5.2 × 106 mins 
+ 
+ 
+ 
+= 8.66 × 104 hours = 3611 days 
+Total Motorcycle ride reduce: 0.5 ×  724800 × 15 km = 5.4 × 106 km 
+Time saving for Motorcycle: 5.4 × 106  × (8.57 – 1.5) mins = 3.8 × 107 mins 
+ 
+ 
+ 
+= 6.3 × 105 hours = 26289 days 
+Total working time saving: 50000 + 720 + 3611 + 26289 days = 80620 days 
+4.3.2.2 Fuel and Cost Saving 
+The cost of installation or repair is not listed here rather we considering only running 
+cost. Because whenever users switch from car to bicycle there would be a significant 
+reduction in costs. 
+ 
+Here it is assumed that the mileage is 5km per liter diesel for bus and 20km per liter 
+diesel for both car and taxicab of 65tk per liter and the mileage is estimated to be 50km 
+per liter for motorcycle of 89tk per liter. On the other hand, the electric cycle charges 
+of 10tk with 50km of travel per charge charges. The fuel consumption is calculated per 
+day basis and user saving is grand saving of considering all users of the Dhaka city.  
+ 
+In the following Table 4.12, the needed fuel for cars, taxicabs, motorcycles, and buses 
+is calculated and there is no running cost & fuel cost for the bicycle. In case of bus, the 
+per km fare is fixed by BRTA of 1.7tk and we assumed that taxicab fare is 50tk per km. 
+Here the distance of all the node point is measured from center node point 7. 
+ 
+
+ 
+ 
+43 
+ 
+The gross fuel saving and user cost saving of the entire Dhaka city is described in the 
+following Table 4.13. So, it is saved around 195570 liters fuel and 46.72 million user 
+costs per day for using bicycle in the entire Dhaka area. 
+ 
+Fuel and Cost Saving Explanation: 
+Fuel saving for Bus: 254796 km × 1/5 litres = 50959 litres  
+ 
+Table 4.13: Cost saving per day. 
+Transit 
+% of Transit 
+Reduction 
+Transit 
+Reduces 
+Riding 
+Distance (km) 
+Fuel 
+(litres) 
+User Cost (tk) 
+Bus 
+10% 
+12740 
+254796 
+50959 
+1.27 million 
+Car 
+5% 
+14663 
+146630 
+7331 
+4.5 lakh 
+Taxicab 
+20% 
+7320 
+732000 
+29280 
+36.5 million 
+Motor cycle 
+50% 
+362400 
+5436000 
+1.08 × 105 
+8.5 million 
+Total cost saving  
+195570 
+46.72 million 
+ 
+ 
+Table 4.12: Cost comparison in car, bus and bicycle considering from node point 7. 
+Node 
+Point 
+Dist
+ance 
+(km) 
+Car 
+Taxicabs 
+Motor Cycle 
+Bus 
+Electric 
+Bicycle  
+Fuel 
+(litre) 
+User 
+Cost 
+(tk) 
+Fuel 
+(litre) 
+User 
+Cost 
+(tk) 
+Fuel 
+(litre) 
+User 
+Cost 
+(tk) 
+Fuel 
+(litre) 
+User 
+Cost 
+(tk) 
+Dista
+nce 
+(km) 
+User 
+Cost 
+(tk) 
+1 
+14 
+0.7 
+45.5 
+0.56 
+700 
+0.28 
+15.49 
+ 
+ 
+ 
+ 
+ 
+ 
+As we 
+assume 
+bicycles 
+reduces 
+10% of the 
+total bus in 
+Dhaka 
+city. 
+ 
+The fuel 
+consumpti
+on is 
+reduced  
+= 
+12740×20
+×1/5 litres 
+ 
+= 50959 
+litres 
+23.8 
+15.65 
+3.13 
+2 
+7.5 
+0.38 
+24.38 
+0.3 
+375 
+0.15 
+10.68 
+12.75 
+9.89 
+1.98 
+3 
+18.2 
+0.91 
+59.15 
+0.73 
+910 
+0.36 
+8.37 
+30.94 
+19.51 
+3.9 
+4 
+10.2 
+0.51 
+33.15 
+0.41 
+510 
+0.2 
+6.23 
+17.34 
+9.02 
+1.8 
+5 
+8.7 
+0.44 
+28.28 
+0.35 
+435 
+0.17 
+9.79 
+14.79 
+11.9 
+2.38 
+6 
+6 
+0.3 
+19.5 
+0.24 
+300 
+0.12 
+16.2 
+10.2 
+3.95 
+0.79 
+8 
+4.7 
+0.24 
+15.28 
+0.19 
+235 
+0.09 
+21.36 
+7.99 
+3.17 
+0.63 
+9 
+3.5 
+0.18 
+11.38 
+0.14 
+175 
+0.07 
+17.27 
+5.95 
+2.17 
+0.43 
+10 
+5.5 
+0.28 
+17.88 
+0.22 
+275 
+0.11 
+14.77 
+9.35 
+5.98 
+1.2 
+11 
+9.1 
+0.46 
+29.58 
+0.36 
+455 
+0.18 
+15.49 
+15.47 
+10.61 
+2.12 
+12 
+12 
+0.6 
+39 
+0.48 
+600 
+0.24 
+14.24 
+20.4 
+9.04 
+1.81 
+13 
+9.7 
+0.49 
+31.53 
+0.39 
+485 
+0.19 
+29.55 
+16.49 
+9.44 
+1.89 
+14 
+8.3 
+0.42 
+26.98 
+0.33 
+415 
+0.17 
+20.29 
+14.11 
+7.31 
+1.46 
+15 
+8.7 
+0.44 
+28.28 
+0.35 
+435 
+0.17 
+15.49 
+14.79 
+11.55 
+2.31 
+16 
+8 
+0.4 
+26 
+0.32 
+400 
+0.16 
+3.92 
+13.6 
+8.7 
+1.74 
+17 
+16.6 
+0.83 
+53.95 
+0.66 
+830 
+0.33 
+4.09 
+28.22 
+15.61 
+3.12 
+18 
+11.4 
+0.57 
+37.05 
+0.46 
+570 
+0.23 
+12.1 
+19.38 
+14.78 
+2.96 
+19 
+8.7 
+0.44 
+28.28 
+0.35 
+435 
+0.17 
+11.75 
+14.79 
+9.87 
+1.97 
+20 
+2.2 
+0.11 
+7.15 
+0.09 
+110 
+0.04 
+15.49 
+3.74 
+2.12 
+0.42 
+21 
+2.3 
+0.12 
+7.48 
+0.09 
+115 
+0.05 
+28.3 
+3.91 
+1.76 
+0.35 
+22 
+6.8 
+0.34 
+22.1 
+0.27 
+340 
+0.14 
+26.17 
+11.56 
+7.34 
+1.47 
+23 
+6.6 
+0.33 
+21.45 
+0.26 
+330 
+0.13 
+15.13 
+11.22 
+4.07 
+0.81 
+24 
+8.7 
+0.44 
+28.28 
+0.35 
+435 
+0.17 
+18.16 
+14.79 
+8.55 
+1.71 
+25 
+15.9 
+0.8 
+51.68 
+0.64 
+795 
+0.32 
+34 
+27.03 
+18.72 
+3.74 
+26 
+14.7 
+0.74 
+47.78 
+0.59 
+735 
+0.29 
+15.49 
+24.99 
+15.5 
+3.1 
+27 
+8.5 
+0.43 
+27.63 
+0.34 
+425 
+0.17 
+10.68 
+14.45 
+11.2 
+2.24 
+28 
+10.2 
+0.51 
+33.15 
+0.41 
+510 
+0.2 
+8.37 
+17.34 
+9.41 
+1.88 
+29 
+19.1 
+0.96 
+62.08 
+0.76 
+955 
+0.38 
+6.23 
+32.47 
+22.02 
+4.4 
+ 
+ 
+ 
+
+ 
+ 
+44 
+ 
+User money saving: 254796 km × (
+65
+5 −  40 ×
+10
+50) tk = 1.27 million tk 
+Fuel saving for Car: 146634 km × 
+1
+20 litres = 7331 litres 
+User money saving: 146634 km × (
+65
+20 − 
+10
+50) tk = 4.5 × 105 tk = 4.5 lakh tk 
+Fuel saving for Taxicab: 732000 km × 
+1
+25 litres = 29280 litres 
+User money saving: 732000 km × (50 − 
+10
+50) tk = 3.65 × 107 tk = 36.5 million tk 
+Fuel saving for Motor cycle: 5.4 × 106 km × 
+1
+50 litres = 1.1 × 105 litres  
+User money saving 5.4 × 106 × (
+89
+50 −  
+10
+50) tk = 8.5 × 106 tk = 8.5 million tk 
+Total fuel saving: 50959 + 7331 + 29280 + 1.1 × 105 litres = 197570 litres 
+Total User money saving: 1.27 + .45 + 36.5 + 8.5 million tk = 46.72 million tk 
+4.3.2.3 CO2 Emission Reduction 
+ In the following calculation, 887 g/km, 258 g/km, 237 g/km and 40 g/km are the 
+considered amount of CO2 emission in 1km ride of bus, car, taxicab, and motorcycle 
+respectively. Then CO2 emission is reduced in a significant amount. For the entire 
+Dhaka city, the gross CO2 emission reduction is described in the Table 4.14. In total, 
+around 6.58 × 105 kg CO2 emission is reduced in the entire Dhaka city per day.   
+ 
+CO2 Emission Reduction Explanation: 
+CO2 emission reduction for Bus: 0.1 × 127398 × 20 km × 887 gm = 2.3 × 105 kg 
+CO2 emission reduction for Car: 0.05 × 293268 × 10 km × 258 gm = 3.8 × 104 kg 
+CO2 emission reduction for Taxicab: 0.2 × 36600 × 100 km × 237 gm = 1.7 × 105 kg 
+CO2 emission reduction for Motorcycle: 0.5 ×724800 ×15 km × 40 gm = 2.2 × 105 kg 
+Total CO2 emission reduction: 6.58 × 105 kg 
+ 
+Table 4.14: CO2 emission reduction per day. 
+Transit 
+% of Transit 
+Reduction 
+Transit 
+Reduces 
+Riding 
+Distance (km) 
+CO2 
+Emission (g) 
+Bus 
+10% 
+12740 
+254796 
+2.3 × 108 
+Car 
+5% 
+14663 
+146630 
+3.8 × 107 
+Taxicab 
+20% 
+7320 
+732000 
+1.7 × 108 
+Motor cycle 
+50% 
+362400 
+5436000 
+2.2 × 108 
+Total CO2 emission reduction  
+6.58 × 108 
+ 
+ 
+
+ 
+ 
+45 
+ 
+When CNG is being used as fuel in buses, cars, and taxicabs the CO2 emission is more 
+significant. Buying a car and motorcycle also increases traffic jams, air carbon dioxide, 
+pollution of the environment and costly in the current situation in Dhaka. Alternatively, 
+electric bicycle does not make an enormous traffic jam, CO2 in air, and less expensive. 
+In order to do this, governments should build parking lots in various places where 
+necessary. 
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+ 
+CHAPTER 5 
+Conclusions 
+A modified Physarum-inspired model is presented in this paper to address the design 
+of the bicycle lane network. Different approaches, like exact approaches and heuristic 
+approaches, have been presented over the past decades to design transportation 
+networks. Recently bio-inspired method had drawn great attraction to network design. 
+In real two-way traffic networks, the modified technique is more effective and efficient. 
+This chapter will now give a short summary of the main points described in this thesis. 
+Also, it discusses possible future works based on the outcome of the present work 
+ 5.1 Achievements 
+The network design technology inspired by Physarum is believed to have balanced 
+costs, effectiveness, and resilience. Inside Dhaka city, an unorganized and unplanned 
+city, we have developed an electric bicycle network system where there's little footway. 
+To meet this challenge, we primarily use local roads and try to avoid major roads 
+towards the construction of the electric bicycle network. Since bicycles are non-
+motorized vehicles do not produce greenhouse gases so they do not cause air pollution. 
+They also don't contribute to noise pollution. If a large number of people use bicycle in 
+the city, traffic jams will be eliminated. The costs will be reduced and people can have 
+some physical activity also, which is beneficial to health. Since bicycles do not need to 
+use gasoline, the importation of gasoline will be reduced. That also enriches the 
+economy and the environment. 
+5.2 Future Study 
+In the future, parallel computing and the optimal model for the design of the transport 
+network are part of our work. Furthermore, our research includes the implementation 
+of the Physarum polycephalum inspired model for the dynamic traffic network and the 
+elastic demand traffic network. 
+ 
+
+ 
+ 
+47 
+ 
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+ 
+

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@@ -0,0 +1,1864 @@
+arXiv:2301.02841v1  [math.DS]  7 Jan 2023
+LEVEL-2 LARGE DEVIATION PRINCIPLE FOR
+COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES
+HIROKI TAKAHASI
+Abstract. We consider level-2 large deviations for the one-sided countable full
+shift without assuming the existence of Bowen’s Gibbs state. To deal with non-
+compact closed sets, we provide a sufficient condition in terms of inducing which
+ensures the exponential tightness of a sequence of Borel probability measures
+constructed from periodic configurations. Under this condition we establish the
+level-2 Large Deviation Principle. We apply our results to the continued fraction
+expansion of real numbers in [0, 1) generated by the R´enyi map, and obtain the
+level-2 Large Deviation Principle, as well as a weighted equidistribution of a set
+of quadratic irrationals to equilibrium states of the R´enyi map.
+1. Introduction
+Dynamical systems (iterated maps) equipped with finite Markov partitions are
+represented as finite Markov shifts, and the construction of relevant invariant mea-
+sures and the investigation of their statistical properties are done on the symbolic
+level, with adaptations of ideas in statistical mechanics (see e.g., [4, 5, 29, 32, 39]).
+This thermodynamic formalism initiated in the 60s has been successfully extended
+to maps with infinite Markov partitions and shift spaces with countably infinite
+number of states (see e.g., [1, 6, 10, 11, 18, 30, 31, 41]). This paper is concerned
+with level-2 large deviations for such countable Markov shifts, and its application
+to a dynamical system related to number theory.
+The theory of large deviations aims to characterize limit behaviors of probability
+measures in terms of rate functions. Let X be a topological space, and let M(X )
+denote the space of Borel probability measures on X endowed with the weak*
+topology. We say a sequence {˜µn}∞
+n=1 in M(X ) satisfies the Large Deviation Prin-
+ciple (LDP) if there exists a lower semicontinuous function I : X → [0, ∞] such
+that
+(1.1)
+lim inf
+n→∞
+1
+n log ˜µn(G) ≥ − inf
+G I for any open set G ⊂ X ,
+and
+(1.2)
+lim sup
+n→∞
+1
+n log ˜µn(C) ≤ − inf
+C I for any closed set C ⊂ X .
+We call x ∈ X a minimizer if I(x) = 0 holds. The set of minimizers is a closed
+set. The LDP means that in the limit n → ∞ the measure ˜µn assigns all but
+Date: January 10, 2023.
+2020 Mathematics Subject Classification. 37A44, 37A50, 37A60, 60F10.
+Keywords: thermodynamic formalism; Gibbs state; Large Deviation Principle; periodic points;
+equidistribution.
+1
+
+2
+HIROKI TAKAHASI
+exponentially small mass to the set of minimizers.
+The function I is called a
+rate function, and called a good rate function if its level set {x ∈ X : I(x) ≤
+α} is compact for any α > 0. If X is a metric space and {˜µn}∞
+n=1 satisfies the
+LDP, the rate function is unique. The setup in our mind is that X is the space
+of Borel probability measures on a topological space X on which a Borel map
+σ: X → X acts, and each ˜µn ∈ M(X ) is given in terms of empirical measures
+δn
+x = (1/n) �n−1
+k=0 δσkx, where δσkx ∈ X denotes the unit point mass at σkx ∈ X.
+We refer to the LDP in this setup as level-2 [8, Chapter 1].
+For topologically mixing finite Markov shifts together with H¨older continuous
+potentials, the level-2 LDP for empirical distributions and that for sequences con-
+structed from empirical measures on periodic orbits were established in [15, 22, 33]
+and [16] respectively. A key ingredient in these classical cases is the existence of
+Bowen’s Gibbs states [4]. With the aid of Bowen’s Gibbs states, one can deduce
+the lower bound (1.1) by combining Birkhoff’s and Shannon-McMillan-Breiman’s
+theorems, and the upper bound (1.2) by modifying the standard proof of the
+variational principle [40] (see [33]). For countable Markov shifts, Bowen’s Gibbs
+states were constructed under the assumption of a good regularity of potentials
+and a strong connectivity of transition matrices defining the shift spaces (see e.g.,
+[1, 6, 11, 18, 30]). Several level-2 LDPs were established in [34] under the existence
+of Bowen’s Gibbs states.
+It has been realized that not all dynamically relevant invariant probability mea-
+sures correspond to Bowen’s Gibbs states. One of the best known examples is
+the absolutely continuous invariant probability measure of an interval map of
+Manneville-Pomeau type, with finitely many branches and a neutral fixed point.
+Such a measure still retains a weak form of Bowen’s Gibbs state [41], and has
+the weak Gibbsian property in statistical mechanics sense [9, 17, 42]. For a ther-
+modynamic formalism and level-2 large deviations for a class of this map, see
+[12, 28, 41] and [25, 26] respectively. With these historical developments and the
+abundance of interesting dynamical systems modeled by countable Markov shifts
+without Bowen’s Gibbs states (see e.g., [13, 43]), it is important to establish the
+level-2 LDP for countable Markov shifts without assuming the existence of Bowen’s
+Gibbs states.
+A main new difficulty for countable Markov shifts is a treatment of non-compact
+closed sets. We say a sequence {˜µn}∞
+n=1 of Borel probability measures on a non-
+compact space X is exponentially tight if for any L > 0 there exists a compact set
+K ⊂ X such that
+lim sup
+n→∞
+1
+n log ˜µn(X \ K) ≤ −L.
+If {˜µn}∞
+n=1 is exponentially tight, then the upper bound (1.2) for any compact
+closed set implies (1.2) for any closed set which is not necessarily compact, see
+e.g., [7] for details. The proof of the level-2 LDPs in [34] relies on the existence of
+Bowen’s Gibbs states in order to verify the exponential tightness.
+Our strategy for countable Markov shifts without Bowen’s Gibbs states is to use
+inducing to verify the exponential tightness. Inducing is a familiar procedure in
+ergodic theory originally considered in works by Kakutani, Rohlin and others, and
+
+LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES
+3
+was used in the construction of absolutely continuous invariant measures or Gibbs-
+equilibrium states (see e.g., [2, 3, 23, 24]). An inducing scheme we use here is given
+by the first return map to an a priori fixed domain. In terms of this inducing, we
+will formulate a sufficient condition which ensures the exponential tightness for
+the original system.
+A key concept is that of local Gibbs states introduced in
+Section 2.2.
+1.1. Statements of results. Throughout the rest of this paper, let N denote the
+discrete set of positive integers.
+Let X denote the one-sided infinite Cartesian
+product topological space of N, called a countable full shift. The topology of X
+has a base that consists of cylinders
+[p1 · · · pn] = {x = (xn)∞
+n=1 ∈ X : xk = pk for every k ∈ {1, 2, . . . , n}},
+where n ≥ 1 and p1 · · ·pn ∈ Nn.
+This topology is metrizable with the metric
+d(x, y) = exp (− inf{n ≥ 1: xn ̸= yn}) where exp(−∞) = 0 by convention. Let σ
+denote the left shift acting on X: (σx)n = xn+1 for n ≥ 1.
+Let φ: X → R be a function, called a potential. We say φ is acceptable if it is
+uniformly continuous and satisfies
+sup
+p∈N
+�
+sup
+[p]
+φ − inf
+[p] φ
+�
+< ∞.
+We say φ is locally H¨older continuous if there exist C > 0 and α ∈ (0, 1] such that
+for any p ∈ N and all x, y ∈ [p],
+|φ(x) − φ(y)| ≤ Cd(x, y)α.
+Clearly, if φ is locally H¨older continuous then it is acceptable. For each n ≥ 1 we
+write Snφ for the Birkhoff sum �n−1
+k=0 φ ◦ σk, and introduce a pressure
+(1.3)
+P(φ) = lim
+n→∞
+1
+n log
+�
+p1···pn∈Nn
+sup
+[p1···pn]
+exp Snφ.
+This limit exists by the sub-additivity [4, Lemma 1.18], which is never −∞.
+Let φ: X → R be acceptable and satisfy P(φ) < ∞. We consider a sequence
+{˜µn}∞
+n=1 of Borel probability measures on M(X) given by
+(1.4)
+˜µn =
+1
+Zn(φ)
+�
+x∈En
+exp Snφ(x)δδn
+x,
+where
+En = {x ∈ X : σnx = x},
+and δδn
+x denotes the unit point mass at δn
+x, and Zn(φ) the normalizing constant.
+In dynamical systems terms, En is the set of periodic points of period n.
+In
+statistical mechanics terms, the measure ˜µn is closely related to the canonical
+ensemble subject to a periodic boundary condition.
+An inducing scheme consists of a subset X∗ of X of the form
+(1.5)
+X∗ = X \
+�
+p∈N∩[1,p∗−1]
+[p],
+
+4
+HIROKI TAKAHASI
+where p∗ ≥ 2, and a function R: X∗ → N ∪ {∞} given by
+(1.6)
+R(x) = inf{n ≥ 1: σnx ∈ X∗}.
+Given an inducing scheme (X∗, R) we define an induced map
+(1.7)
+τ : X∗ ∩
+∞
+�
+k=1
+σ−kX∗ �→ σR(x)x ∈ X∗,
+and an inducing domain
+(1.8)
+Σ =
+∞
+�
+n=0
+τ −n
+�
+X∗ ∩
+∞
+�
+k=1
+σ−kX∗
+�
+.
+In other words, τ is the first return map to X∗ and Σ is the domain on which τ
+can be iterated infinitely many times. We call (Σ, τ|Σ) an induced system. Given
+a potential φ: X → R, we introduce a parametrized family of induced potentials
+Φγ : Σ → R (γ ∈ R) by
+(1.9)
+Φγ(x) = SR(x)φ(x) − γR(x).
+As shown in Section 2.1, the induced system has a countably infinite partition that
+conjugates the system to the countable full shift. The local H¨older continuity of
+the induced potential Φγ and its pressure P(Φγ) are well-defined in terms of this
+conjugacy. Our main result is stated as follows.
+Theorem A (the level-2 Large Deviation Principle). Let φ: X → R be acceptable
+and satisfy P(φ) < ∞.
+Assume there exists an induced system for which the
+induced potentials Φγ, γ ∈ R are locally H¨older continuous, and there exists γ0 ∈ R
+such that P(Φγ0) = 0. Then {˜µn}∞
+n=1 is exponentially tight and satisfies the LDP
+with the good rate function.
+Let us define the rate function in Theorem A. Let M(X, σ) denote the set of σ-
+invariant elements of M(X) and let Mφ(X, σ) = {µ ∈ M(X, σ):
+�
+φdµ > −∞}.
+Define Fφ : M(X) → [−∞, 0] by
+Fφ(µ) =
+�
+h(µ) +
+�
+φdµ − P(φ)
+if µ ∈ Mφ(X, σ);
+−∞
+otherwise,
+where h(µ) ∈ [0, ∞] denotes the measure-theoretic entropy of µ with respect to σ.
+Since φ is acceptable and P(φ) < ∞, sup φ < ∞ is finite. For each µ ∈ Mφ(X, σ),
+�
+φdµ is finite [18, Theorem 2.1.9] and we have h(µ) +
+�
+φdµ ≤ P(φ) < ∞, and so
+h(µ) < ∞. If φ is acceptable, then we have
+P(φ) = sup
+�
+h(µ) +
+�
+φdµ: µ ∈ Mφ(X, σ)
+�
+,
+known as the variational principle [18, Theorem 2.1.8]. A measure µ ∈ Mφ(X, σ)
+which attains this supremum is called an equilibrium state for the potential φ. The
+rate function Iφ : M(X) → [0, ∞] in Theorem A is given by
+(1.10)
+Iφ(µ) = − inf
+G∋µ sup
+G
+Fφ,
+
+LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES
+5
+where the infimum is taken over all open subsets G of M(X) containing µ. Since
+the entropy is not upper semicontinuous on M(X, σ), Iφ may not be equal to −Fφ.
+Since the sequence {˜µn}∞
+n=1 in Theorem A is exponentially tight, it is tight. By
+Prohorov’s theorem, it has a limit point. Since the rate function Iφ in Theorem A
+is the good rate function, there exists at least one minimizer. If the minimizer
+is unique, we obtain a “level-2 weighted equidistribution of elements of �∞
+n=1 En
+toward minimizers”.
+Theorem B (level-2 weighted equidistribution). Let φ: X → R be acceptable and
+satisfy P(φ) < ∞. Assume there exists an induced system for which the induced
+potentials Φγ, γ ∈ R are locally H¨older continuous, and there exists γ0 ∈ R such
+that P(Φγ0) = 0. Assume that the minimizer of the rate function Iφ is unique,
+denoted by µmin. For any bounded continuous function ˜ϕ: M(X) → R,
+lim
+n→∞
+1
+Zn(φ)
+�
+x∈En
+exp Snφ(x) ˜ϕ(δn
+x) = ˜ϕ(µmin).
+Under the assumption of Theorem A, minimizers are not always unique, and not
+always an equilibrium state. A sufficient condition was given in [35] which ensures
+that minimizers are equilibrium states.
+Taking various continuous functions ˜ϕ in Theorem B, we obtain convergences
+of various time averages over the elements of En. Let C(X) denote the set of
+real-valued bounded continuous functions on X.
+Corollary (Inspired by Olsen [21, Section 1.1]). Under the assumption of Theo-
+rem B, assume moreover the minimizer is unique, denoted by µmin.
+(a) For all ϕ, ψ ∈ C(X),
+lim
+n→∞
+1
+Zn(φ)
+�
+x∈En
+exp Snφ(x) 1
+n2Snϕ(x)Snψ(x) =
+�
+ϕdµmin
+�
+ψdµmin.
+(b) For ϕ, ψ ∈ C(X) with inf ψ > 0,
+lim
+n→∞
+1
+Zn(φ)
+�
+x∈En
+exp Snφ(x)Snϕ(x)
+Snψ(x) =
+�
+ϕdµmin
+�
+ψdµmin
+.
+(c) For π1, π2 ∈ C(X) and a bounded continuous function f : R → R,
+lim
+n→∞
+1
+Zn(φ)
+�
+x∈En
+exp Snφ(x) 1
+n2
+n−1
+�
+k1,k2=0
+f(π1(σk1x) + π2(σk2x))
+=
+�
+fd(µmin ◦ π−1
+1
+⊗ µmin ◦ π−1
+2 ),
+where ⊗ denotes the convolution.
+Proof. Apply Theorem B to the bounded continuous functions µ ∈ M(X) �→
+�
+ϕdµ
+�
+ψdµ, µ ∈ M(X) �→
+�
+ϕdµ/
+�
+ψdµ, µ ∈ M(X) �→
+�
+fd(µ ◦ π−1
+1
+⊗ µ ◦ π−1
+2 )
+respectively.
+□
+
+6
+HIROKI TAKAHASI
+ 1
+0
+1/2 2/3
+1
+Figure 1. The graph of the R´enyi map T.
+1.2. Applications. Our results can be applied to dynamical systems modeled by
+the countable full shift without Bowen’s Gibbs state. The assumption in Theo-
+rem A can be verified, for example, for the infinite Manneville-Pomeau map [13,
+Section 2.2], and the two-dimensional conformal maps in [43, Section 5] related to
+number theory. Minimizers of the associated rate functions are not unique, and so
+Theorem B does not apply. Further applications of different taste will be given in
+our forthcoming paper.
+A prime example to which our results apply is the R´enyi map T : [0, 1) → [0, 1)
+given by
+(1.11)
+T(ξ) =
+1
+1 − ξ −
+�
+1
+1 − ξ
+�
+,
+where ⌊·⌋ denotes the floor function. The graph of T is obtained by reversing the
+graph of the well-known Gauss map ξ ∈ (0, 1] → 1/ξ − ⌊1/ξ⌋ ∈ [0, 1) around the
+axis {ξ = 1/2}, as shown in FIGURE 1. The map T leaves invariant the absolutely
+continuous infinite measure dx/x, and x = 0 is its neutral fixed point:T(0) = 0,
+T ′(0) = 1. The asymptotic distribution of typical orbits, in the Lebesgue measure
+sense, are concentrated on this neutral fixed point.
+The iteration of T generates an infinite continued fraction expansion of each
+number ξ ∈ [0, 1) of the form
+(1.12)
+ξ = 1 −
+1
+d1(ξ) −
+1
+d2(ξ) − ...
+,
+where dn(ξ) = ⌊1/(1 − T n−1(ξ))⌋ + 1 ≥ 2 for n ≥ 1. Using the infinite Markov
+partition {Jp}p∈N, Jp = [1 − 1/p, 1 − 1/(p + 1)) of [0, 1), one can represent T as
+the left shift acting on X [13, 37]. The map
+(1.13)
+π: (xn)∞
+n=1 ∈ X �→ π((xn)∞
+n=1) ∈
+∞
+�
+n=1
+T −n+1(J(xn)) ⊂ [0, 1)
+is a well-defined homeomorphism onto its image satisfying T ◦ π = π ◦ σ. We
+consider the potential φ = − log |T ′ ◦ π|, where T ′ denotes the derivative of T
+which is one-sided at boundary points of the Markov partition. From the mean
+
+LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES
+7
+value theorem applied to the inverse branches of T, for any p ≥ 1 and all ξ, η ∈ Jp
+we have
+log |T ′(ξ)|
+|T ′(η)| ≤ 2|T(ξ) − T(η)| < 2.
+In particular, φ is acceptable. Since sup[p] eφ is comparable to p−2, P(βφ) < ∞
+holds if and only if �
+p∈N p−2β is finite, which is equivalent to β > 1/2. It is easy
+to see that for n ≥ 1, T n maps [0, 1/(n + 1)) diffeomorphically onto [0, 1). The
+mean value theorem implies limn→∞ sup[0,1/(n+1)) |(T n)′| = ∞, while |(T n)′(0)| = 1
+for n ≥ 1. It follows that for any β > 1/2 there is no Bowen’s Gibbs state for
+the potential βφ. Meanwhile, it is known [13] that for β > 1/2, the equilibrium
+state for βφ is unique, which we denote by µβφ. For 1/2 < β < 1, µβφ has positive
+entropy and fully supported. For β ≥ 1, µβφ is the unit point mass at π−1(0).
+Let I denote the set of irrational numbers in (0, 1). The set En corresponds to
+the set of numbers in I ∪ {0} for which the continued fraction (1.12) is periodic
+of period n. As in the proof of [14, Theorem 28], one can show that any number
+in �∞
+n=1{ξ ∈ I: T n(ξ) = ξ} is a quadratic irrational, i.e., an irrational root of a
+quadratic polynomial with integer coefficients. Conversely, any quadratic irrational
+in I has an eventually periodic continued fraction of the form (1.12), see [20,
+Theorem 3].
+An induced system as in Theorem A is obtained from the first return map to
+the interval (1/2, 1) not containing the neutral fixed point. From Theorems A and
+B we obtain the following. For ξ ∈ [0, 1) and n ≥ 1, let δn
+ξ denote the empirical
+measure (1/n) �n−1
+k=0 δT k(ξ) on [0, 1).
+Theorem C. For any 1/2 < β ≤ 1, the sequence of Borel probability measures on
+M(π(X)) given by
+1
+Zn(βφ)
+�
+ξ∈I∪{0}
+T n(ξ)=ξ
+|(T n)′(ξ)|−βδδn
+ξ
+for n = 1, 2, . . .
+satisfies the LDP. The minimizer is unique and it is the unit point mass at µβφ◦π−1.
+Moreover, for any bounded continuous function ˜ϕ: M(π(X)) → R we have
+lim
+n→∞
+1
+Zn(βφ)
+�
+ξ∈I∪{0}
+T n(ξ)=ξ
+|(T n)′(ξ)|−β ˜ϕ(δn
+ξ ) = ˜ϕ(µβφ ◦ π−1).
+1.3. Structure of the paper. The rest of this paper consists of two sections. In
+Section 2 we verify the exponential tightness of the sequence in (1.4) under the
+assumption of Theorem A. In Section 3 we complete proofs of all the theorems.
+We close with a remark on possible generalizations of the main results.
+2. Exponential tightness
+The aim of this section is to verify the exponential tightness of the sequence in
+(1.4). In Section 2.1 we start with a symbolic representation of the induced system.
+In Section 2.2 we introduce the notion of local Gibbs states. In Section 2.3 we prove
+a main technical estimate assuming the existence of a local Gibbs state. Using this
+
+8
+HIROKI TAKAHASI
+estimate, we verify the exponential tightness in Section 2.4. In Section 2.5 we show
+that the assumption of Theorem A implies the existence of a local Gibbs state.
+2.1. Symbolic representation of the induced system. For a set S and an
+integer j ≥ 1, let Sj denote the set of words of elements of S of word length j. We
+introduce an empty word ∅ and set S0 = {∅}, a∅ = a = ∅a, a∅b = ab for a, b ∈ S.
+We set W(S) = �
+j≥1 Sj, N0 = N ∪ {0} and W0(S) = W(S) ∪ S0.
+Let (X∗, R) be an inducing scheme. Let
+N∗ = N ∩ [p∗, ∞) and N∗ = N ∩ [1, p∗ − 1).
+For each p ∈ N∗ and ω ∈ W0(N∗), the set �
+q∈N∗[pωq] is mapped by the induced
+map τ in (1.7) bijectively onto X∗. Since the domain X∗∩�∞
+k=1 σ−kX∗ of definition
+of τ is partitioned into countably infinite sets of this form, the induced system τ|Σ
+is represented as the countable full shift over the infinite alphabet
+(2.1)
+A =
+� �
+q∈N∗
+[pωq]: p ∈ N∗ and ω ∈ W0(N∗)
+�
+.
+To make this statement into a rigorous one, we endow A with the discrete
+topology, and consider the countable full shift
+AN = {z = (zn)∞
+n=1: zn ∈ A for every n ≥ 1}.
+We use bold letters to denote elements of W(A), and a double square bracket [[·]]
+to denote cylinders in AN: the n-cylinder (n ≥ 1) spanned by a = a1 · · · an ∈ An is
+[[a]] = {z = (zk)∞
+k=1 ∈ AN : zk = ak for every k ∈ {1, . . . , n}}.
+By definition, for each k ∈ {1, . . . , n} we have ak = �
+q∈N∗[pkωjkq] where pk ∈ N∗,
+jk ∈ N0, ωjk ∈ Njk
+∗ . Let ∥a∥ denote the word length of p1ωj1p2ωj2 · · · pnωjn in
+W(N), namely
+∥a∥ = n + j1 + · · · + jn.
+Let [a] denote the corresponding ∥a∥-cylinder in X, namely
+[a] = [p1ωj1p2ωj2 · · · pnωjn] ⊂ X.
+It is easy to check that a coding map Π: AN → Σ given by
+(2.2)
+Π: (zn)∞
+n=1 ∈ AN �→ Π((zn)∞
+n=1) ∈
+∞
+�
+n=1
+[z1 · · · zn] ⊂ Σ
+is well-defined, and is a homeomorphism. Let θ denote the left shift acting on AN.
+Clearly we have Π ◦ θ = τ|Σ ◦ Π.
+The following notation will be frequently used later. For a = a1 · · · an ∈ An as
+above with [a] = [p1ωj1p2ωj2 · · · pnωjn] and q ∈ N∗, let
+aq = p1ωj1p2ωj2 · · · pnωjnq ∈ W(N).
+Lemma 2.1. Let (Σ, τ|Σ) be an induced system and let Π be the coding map in
+(2.2).
+
+LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES
+9
+(a) For every a ∈ W(A),
+Π[[a]] = Σ ∩
+�
+q∈N∗
+[aq].
+(b) If a, b ∈ W(A) satisfy ∥a∥ = ∥b∥, then a = b or [[a]] ∩ [[b]] = ∅.
+Proof. If n ≥ 1, a ∈ An then �n−1
+k=0 R ◦ τ k equals ∥a∥ on Π[[a]], which implies (a).
+If ∥a∥ = ∥b∥ and a ̸= b then (a) implies Π[[a]] ∩ Π[[b]] = ∅, and so [[a]] ∩ [[b]] = ∅,
+which verifies (b).
+□
+2.2. Local Gibbs states. Let φ: X → R satisfy P(φ) < ∞ and let (Σ, τ|Σ) be
+an induced system. A Borel probability measure λφ on AN is called a local Gibbs
+state for the potential φ associated with (Σ, τ|Σ), if there exist constants C ≥ 1,
+γ0 ∈ R such that for any a ∈ W(A) and any x ∈ Π[[a]] we have
+(2.3)
+C−1 ≤
+λφ[[a]]
+exp
+�
+S∥a∥φ(x) − γ0∥a∥
+� ≤ C.
+We do not require the θ-invariance. If the context is clear, we simply call λφ a
+local Gibbs state (associated with (Σ, τ|Σ)).
+If λφ is a local Gibbs state, then for any a ∈ W(A), the λφ-measure of the
+cylinder [[a]] in AN is given (up to multiplicative constants) by the Birkhoff sum of
+φ along the orbit of x of length ∥a∥ and the word length ∥a∥. Since the word length
+of a as a word in W(A) does not appear in the formula (2.3), λφ well captures part
+of the original dynamics (X, σ).
+If λφ is a local Gibbs state, the Borel probability measure λφ ◦ Π−1 on Σ is
+τ|Σ-invariant. These two measures are related as follows.
+Lemma 2.2. Let φ: X → R satisfy P(φ) < ∞, let (Σ, τ|Σ) be an induced system
+and let λφ be a local Gibbs state associated with (Σ, τ|Σ). For any a ∈ W(A) we
+have
+λφ[[a]] = λφ ◦ Π−1(Σ ∩ [a]).
+Proof. We write νφ for λφ ◦ Π−1, and {R = n} for {z ∈ Σ: R(z) = n} for each
+n ≥ 1. We have νφ{R = n} > 0 for every n ≥ 1. Let νφ|{R=n} denote the restriction
+of νφ to {R = n}. The measure
+µφ =
+∞
+�
+n=1
+n−1
+�
+k=0
+νφ|{R=n} ◦ σ−k
+is a finite measure if and only if
+�
+Rdνφ < ∞. Since {R = n} is disjoint from
+�n−1
+k=1 σ−k(Σ) for n ≥ 2, we have
+(2.4)
+µφ|Σ =
+∞
+�
+n=1
+νφ|{R=n} = νφ = λφ ◦ Π−1.
+For any a ∈ W(A) we have Π[[a]] ⊂ Σ, and so
+(2.5)
+λφ[[a]] = µφΠ[[a]].
+
+10
+HIROKI TAKAHASI
+By µφ|Σ = νφ in (2.4) and Lemma 2.1(a), for any a ∈ W(A) we have
+(2.6)
+µφΠ[[a]] = νφΠ[[a]] =
+�
+q∈N∗
+νφ(Σ ∩ [aq]).
+Let j ≥ 1 be such that a ∈ Aj. Then σ∥a∥ and τ j coincide on Σ ∩ [a]. Since
+�
+q∈N∗(Σ∩[aq]) ⊂ (τ|Σ)−j(�
+q∈N∗[q]) = ∅ by τ(Σ) ⊂ Σ, we have νφ(�
+q∈N∗ Σ∩[aq]) =
+0. Combining this with (2.5), (2.6) we obtain
+λφ[[a]] =
+�
+q∈N∗
+νφ(Σ ∩ [aq]) +
+�
+q∈N∗
+νφ(Σ ∩ [aq]) = νφ(Σ ∩ [a]),
+as required.
+□
+2.3. Exponential decay on partition functions. The next proposition pro-
+vides a main technical estimate under the existence of a local Gibbs state.
+Proposition 2.3. Let φ: X → R be acceptable and satisfy P(φ) < ∞. Assume
+there exist an induced system (Σ, τ|Σ) and an associated local Gibbs state λφ. There
+exist δ′ ∈ (0, 1/5] and n0 ≥ 1 such that if δ ∈ (0, δ′] and {Ni}∞
+i=1 is a non-decreasing
+integer sequence such that
+(2.7)
+max N∗ ≤ N1 and
+(2.8)
+∞
+�
+k=Ni+1
+�
+a∈A
+Π[[a]]⊂[k]
+λφ[[a]] ≤ δ2i for every i ≥ 1,
+then for every n ≥ n0 and every m ∈ {1, . . . , n} we have
+(2.9)
+�
+x∈En
+δn
+x (X\Γ)=m/n
+exp Snφ(x) ≤ eγ0n2nn(4δ)m
+1 − 4δ ,
+where
+(2.10)
+Γ = {x = (xi)∞
+i=1 ∈ X : xi ≤ Ni for every i ≥ 1}.
+Proposition 2.3 asserts that contributions of elements of En to Zn(φ) whose orbit
+escape from the compact set Γ exactly m times within period n is exponentially
+small in m. Similar estimates were obtained in [34] under the existence of Bowen’s
+Gibbs states.
+Proof of Proposition 2.3. Since λφ is a local Gibbs state, there exist constants C ≥
+1 and γ0 ∈ R such that for any a ∈ W(A) and any x ∈ Π[[a]] we have
+(2.11)
+C−1 ≤
+λφ[[a]]
+exp
+�
+S∥a∥φ(x) − γ0∥a∥
+� ≤ C.
+For the rest of the proof of Proposition 2.3, we shall use the notation a ≪ b for
+two positive reals a, b to indicate that a/b is bounded from infinity by a constant
+which depends only on C. If a ≪ b and b ≪ a, we shall write a ≍ b.
+
+LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES
+11
+The first inequality in (2.11) will be used to bound a partial sum of Zn(φ) from
+above by a sum of λφ-measures of cylinders in AN. Further, we will bound this
+sum using the product property which is a consequence of (2.11):
+(2.12)
+λφ[[ab]] ≍ λφ[[a]]λφ[[b]] for a, b ∈ W(A).
+Let δ ∈ (0, 1/5] and let {Ni}∞
+i=1 be a non-decreasing integer sequence satisfying
+(2.7) and (2.8). Let Γ = Γ({Ni}∞
+i=1) be the compact subset of X given by (2.10).
+If x ∈ En \ �n−1
+i=0 σ−i(Σ) then xi ≤ max N∗ = N1 ≤ Ni for 1 ≤ i ≤ n, and so
+δn
+x(Γ) = 1. By this and the periodicity of elements of En, for 1 ≤ m ≤ n we have
+�
+x∈En
+δn
+x (X\Γ)=m/n
+exp Snφ(x) =
+�
+x∈En∩�n−1
+i=0 σ−i(Σ)
+δn
+x (X\Γ)=m/n
+exp Snφ(x)
+≤ n
+�
+x∈En∩Σ
+δn
+x (X\Γ)=m/n
+exp Snφ(x).
+(2.13)
+To bound the last sum in (2.13), we decompose the set {x ∈ En ∩ Σ: δn
+x(X \
+Γ) = m/n} into subsets sharing the same itinerary up to time n, and estimate
+a contribution from each subset separately, and finally unify all these estimates
+counting the total number of possible itineraries.
+Define a function r: X \ Γ → N by
+r(x) = min{i ≥ 1: xi > Ni}.
+From (2.7) we have Σ ⊂ X \ Γ. Hence, for each x ∈ Σ there are infinitely many
+i ≥ 0 with σix /∈ Γ. By an itinerary of x ∈ Σ we mean two sequences {nj(x)}∞
+j=1,
+{rj(x)}∞
+j=1 in N0 given by the recursion formulas
+n1(x) = min{i ≥ 0: σix /∈ Γ}, and
+rj(x) = r(σnj(x)x), nj+1(x) = min{i ≥ nj(x) + rj(x): σix /∈ Γ} for j ≥ 1.
+Lemma 2.4. Let x ∈ Σ.
+(a) {i ≥ 0: σix /∈ Γ} = �∞
+j=1[nj(x), nj(x) + rj(x) − 1] ∩ N0.
+(b) xnj(x)+rj(x) ∈ N∗ for every j ≥ 1.
+Proof. Since {Ni}∞
+i=1 is non-decreasing, if x /∈ Γ then σix /∈ Γ for 0 ≤ i ≤ r(x) −
+1.
+This implies (a).
+Since σnj(x)x = xnj(x)+1xnj(x)+2 · · · and σnj(x)x /∈ Γ with
+r(σnj(x)x) = rj(x), we obtain xnj(x)+rj(x) > Nrj(x) ≥ N1, which together with (2.7)
+yields xnj(x)+rj(x) ∈ N∗ as in (b).
+□
+For each j ∈ {1, . . . , m} and n1 · · · nj ∈ Nj
+0, r1 · · · rj ∈ Nj with n1 < · · · < nj ≤
+n, we put
+(2.14)
+∆
+r1···rj
+n1···nj = {x ∈ En ∩ Σ: (ni(x), ri(x)) = (ni, ri) for every i ∈ {1, . . . , j}}.
+Lemma 2.5. If δ > 0 is sufficiently small, then for j ∈ {1, . . . , m}, n1 · · · nj ∈ Nj
+0
+and r1 · · · rj ∈ Nj such that ∆
+r1···rj
+n1···nj ̸= ∅ we have
+�
+x∈∆
+r1···rj
+n1···nj
+exp Snφ(x) ≤ eγ0nδr1+···+rj.
+
+12
+HIROKI TAKAHASI
+Proof. We start with the case j = 1. We introduce two sets of induced words
+B0 = {b ∈ W(A): ∥b∥ = n − n1 − r1 + 1, Π[[b]] ⊂ ∪∞
+k=Nr1+1[k]} and
+D0 = {d ∈ W(A): ∥d∥ = n1 + r1 − 1}.
+For each x ∈ ∆r1
+n1 we have xn+1 = x1 ∈ N∗ by the definition (2.14), and xn1+r1 ∈
+N∗ by Lemma 2.4(b). Hence there exist d ∈ D0 and b ∈ B0 such that [d] =
+[x1 · · ·xn1+r1−1] and [b] = [xn1+r1 · · · xn], and so x ∈ Π[[db]]. By (2.11) and (2.12),
+exp Snφ(x) ≪ eγ0nλφ[[db]] ≪ eγ0nλφ[[d]]λφ[[b]].
+Summing this inequality over all x ∈ ∆r1
+n1 and then using �
+b∈B0 λφ[[b]] ≤ δ2r1 from
+(2.8) and �
+b∈D0 λφ[[d]] ≤ 1 which follows from Lemma 2.1(b), we obtain
+�
+x∈∆r1
+n1
+exp Snφ(x) ≪eγ0n �
+d∈D0
+λφ[[d]]
+�
+b∈B0
+λφ[[b]] ≤ eγ0nδ2r1 ≤ eγ0nδr1,
+(2.15)
+provided δ is small enough. In case m = 1 we are done.
+To proceed, suppose m ≥ 2. let j, j+1 ∈ {1, . . . , m} and let n1 · · · njnj+1 ∈ Nj+1
+0
+,
+r1 · · · rjrj+1 ∈ Nj+1 be such that ∆
+r1···rjrj+1
+n1···njnj+1 ̸= ∅. Define
+Aj = {a ∈ W(A): ∥a∥ = nj + rj, Π[[a]] ∩ ∆
+r1···rj+1
+n1···nj+1 ̸= ∅},
+Bj = {b ∈ W(A): ∥b∥ = n − nj+1 − rj+1 + 1, Π[[b]] ⊂ ∪∞
+k=Nrj+1+1[k]},
+Cj = {c ∈ W(A): ∥c∥ = n − nj − rj} ,
+Dj = {d ∈ W(A): ∥d∥ = nj+1 + rj+1 − 1} .
+Let a ∈ Aj.
+For each c ∈ Cj we have ∥ac∥ = n.
+Since X is the full shift,
+Π[[ac]] contains a unique element of En ∩ Σ which we denote by ac. Then we have
+ac ∈ Π[[a]] ∩ ∆
+r1···rj
+n1···nj. By (2.11) and (2.12),
+exp Snφ(ac) ≫ eγ0nλφ[[a]]λφ[[c]].
+This implies
+�
+x∈Π[[a]]∩∆
+r1···rj
+n1···nj
+exp Snφ(x) ≫ eγ0nλφ[[a]]
+�
+c∈Cj
+λφ[[c]].
+(2.16)
+By Lemma 2.2, for each c ∈ Cj we have
+λφ[[c]] = λφ ◦ Π−1(Σ ∩ [c]).
+Since the sets Σ ∩ [c], c ∈ Cj are pairwise disjoint and their union equals Σ,
+(2.17)
+�
+c∈Cj
+λφ[[c]] = 1.
+Combining (2.16) and (2.17) yields
+�
+x∈Π[[a]]∩∆
+r1···rj
+n1···nj
+exp Snφ(x) ≫ eγ0nλφ[[a]].
+(2.18)
+Similarly to the case j = 1, for each x ∈ Π[[a]]∩∆
+r1···rj+1
+n1···nj+1 we have xn+1 = x1 ∈ N∗
+by the definition (2.14) and xnj+1+rj+1 ∈ N∗ by Lemma 2.4(b). Hence there exist
+
+LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES
+13
+d ∈ Dj, b ∈ Bj such that [d] = [x1 · · · xnj+1+rj+1−1] and [b] = [xnj+1+rj+1 · · ·xn].
+We have [[d]] ⊂ [[a]] and x ∈ Π[[db]], and by (2.11) and (2.12),
+exp Snφ(x) ≪ eγ0nλφ[[d]]λφ[[b]].
+Summing this inequality over all x ∈ Π[[a]]∩∆
+r1···rj+1
+n1···nj+1, and then using �
+b∈Bj λφ[[b]] ≤
+δ2rj+1 from (2.8) and �
+d∈Dj
+[[d]]⊂[[a]]
+λφ[[d]] ≤ λφ[[a]] from Lemma 2.1(b), we obtain
+�
+x∈Π[[a]]∩∆
+r1···rj+1
+n1···nj+1
+exp Snφ(x) �� eγ0n �
+d∈Dj
+[[d]]⊂[[a]]
+λφ[[d]]
+�
+b∈Bj
+λφ[[b]]
+≤ eγ0nλφ[[a]]δ2rj+1.
+(2.19)
+The two estimates in (2.18) and (2.19) yield
+�
+x∈Π[[a]]∩∆
+r1···rj+1
+n1···nj+1 exp Snφ(x)
+�
+x∈Π[[a]]∩∆
+r1···rj
+n1···nj exp Snφ(x) ≤ δrj+1,
+provided δ is small enough. Rearranging this inequality and summing the result
+over all a ∈ Aj yields
+�
+x∈∆
+r1···rj+1
+n1···nj+1
+exp Snφ(x) ≤ δrj+1
+�
+x∈∆
+r1···rj
+n1···nj
+exp Snφ(x).
+Applying this inequality recursively and combining the final result with (2.15)
+yields the desired inequality in Lemma 2.5.
+□
+For integers L ≥ m and s ∈ {1, . . . , m}, we denote by KL,s the set of elements
+(n1 · · · ns, r1 · · · rs) of Ns
+0×Ns such that 0 ≤ n1 < · · · < ns ≤ n and r1+· · ·+rs = L.
+The number of ways of locating n1, . . . , ns in [0, n] does not exceed ( n
+s ), and for
+each location (n1, . . . , ns) the number of all feasible combinations of (r1, . . . , rs)
+with r1 +· · ·+rs = L is bounded by the number of ways of dividing L objects into
+s groups, not exceeding
+� L+s−1
+s−1
+�
+≤ 2L+s−1. This yields #KL,s ≤ ( n
+s )
+� L+s−1
+s−1
+�
+≤
+2n2L+s−1. Clearly, for each x ∈ En ∩ Σ satisfying δn
+x(X \ Γ) = m/n there exist
+L ≥ m, s ∈ {1, . . . , m} and (n1 · · · ns, r1 · · · rs) ∈ KL,s such that x ∈ ∆r1···rs
+n1···ns. If
+δ ∈ (0, 1/5] is sufficiently small, then together with Lemma 2.5 we obtain
+�
+x∈En∩Σ
+δn
+x (X\Γ)=m/n
+exp Snφ(x) ≤
+m
+�
+s=1
+∞
+�
+L=m
+�
+(n1···ns,r1···rs)∈KL,s
+�
+x∈∆r1···rs
+n1···ns
+exp Snφ(x)
+≤ 2n
+m
+�
+s=1
+∞
+�
+L=m
+2L+s−1δL ≤ 2n
+∞
+�
+L=m
+(4δ)L = 2n(4δ)m
+1 − 4δ .
+From this and (2.13), (2.9) follows and the proof of Proposition 2.3 is complete.
+□
+
+14
+HIROKI TAKAHASI
+2.4. Verifying exponential tightness. We now use Proposition 2.3 to show the
+desired exponential tightness.
+Proposition 2.6. Let φ: X → R be acceptable such that P(φ) < ∞ and let
+(Σ, τ|Σ) be an induced system. If there exists a local Gibbs state for the potential
+φ associated with (Σ, τ|Σ), then {˜µn}∞
+n=1 is exponentially tight.
+Proof. The argument below is an adaptation of the proof of Sanov’s theorem (see
+e.g., [7]) to our setup. For each integer ℓ ≥ 1, we fix δℓ ∈ (0, 1/5] such that
+(2.20)
+1
+1 − 4δℓ
+∞
+�
+m=0
+e2ℓ2m(4δℓ)m ≤ 2.
+We apply Proposition 2.3 and fix a non-decreasing integer sequence {Ni}∞
+i=1 such
+that (2.7) and (2.8) with δ = δℓ hold. We define a compact subset Γℓ = Γ({Ni}∞
+i=1)
+of X by (2.10), and set
+Kℓ =
+�
+ν ∈ M(X): ν(Γℓ) ≥ 1 − 1
+ℓ
+�
+.
+Since M(X) is a Polish space and Γℓ is a closed set, the weak* convergence µk → µ
+for a sequence {µk}∞
+k=1 in Kℓ implies lim supk→∞ µk(Γℓ) ≤ µ(Γℓ). Hence, Kℓ is a
+closed set. For an integer L ≥ 1 we define
+KL =
+∞
+�
+ℓ=L
+Kℓ.
+This set is tight, and by Prohorov’s theorem any sequence in KL has a limit point.
+Hence it is sequentially compact. Since the weak* topology is metrizable with the
+bounded Lipschitz metric, KL is compact. By Chebyshev’s inequality, for n ≥ 1
+we have
+�
+x∈En
+exp(ℓ2nδn
+x (X\Γℓ))≥eℓn
+exp Snφ(x) ≤ e−2ℓn
+�
+x∈En
+δn
+x (X\Γℓ)≥1/n
+exp
+�
+2ℓ2nδn
+x(X \ Γℓ)
+�
+exp Snφ(x)
+= e−2ℓn
+n
+�
+m=1
+e2ℓ2m
+�
+x∈En
+δn
+x (X\Γ)=m/n
+exp Snφ(x).
+Combining this inequality with Proposition 2.3 and (2.20), we have
+�
+x∈En
+δn
+x /∈Kℓ
+exp Snφ(x) =
+�
+x∈En
+δn
+x (X\Γℓ)≥ 1
+ℓ
+exp Snφ(x) =
+�
+x∈En
+exp(ℓ2nδn
+x (X\Γℓ))≥eℓn
+exp Snφ(x)
+≤
+2nn
+1 − 4δℓ
+eγ0ne−2ℓn
+n
+�
+m=0
+e2ℓ2m(4δℓ)m
+≤ 10 · 2nneγ0ne−2ℓn.
+
+LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES
+15
+If L ≥ 1 is large enough, then
+˜µn(M(X) \ KL) ≤
+∞
+�
+ℓ=L
+˜µn(M(X) \ Kℓ) =
+∞
+�
+ℓ=L
+1
+Zn(φ)
+�
+x∈En
+δn
+x /∈Kℓ
+exp Snφ(x)
+≤ 10 · 2nneγ0n
+Zn(φ)
+∞
+�
+ℓ=L
+e−2ℓn ≤ 2neγ0n
+Zn(φ) e−Ln.
+Combining this with the equality limn→∞(1/n) log Zn(φ) = P(φ) < ∞ which fol-
+lows from the uniform continuity of φ, we obtain
+lim sup
+n→∞
+1
+n log ˜µn(M(X) \ KL) ≤ −L + log 2.
+Since L ≥ 1 is an arbitrary large integer, {˜µn}∞
+n=1 is exponentially tight.
+□
+2.5. Existence of a local Gibbs state. The next proposition ensures the exis-
+tence of a local Gibbs state under the assumption of Theorem A.
+Proposition 2.7. Let φ: X → R satisfy P(φ) < ∞. Assume there exists an
+induced system (Σ, τ|Σ) for which the induced potentials Φγ, γ ∈ R associated with
+φ are locally H¨older continuous, and there exists γ0 ∈ R such that P(Φγ0) = 0.
+Then there exists a local Gibbs state for the potential φ associated with (Σ, τ|Σ).
+Proof. Note that P(Φγ0 ◦ Π) = P(Φγ0) = 0. Since AN is the countable full shift,
+the finiteness of P(Φγ0 ◦ Π) implies the summability of the potential Φγ0 ◦ Π. By
+[18, Corollary 2.7.5] together with the summability and the local H¨older continuity
+of Φγ0 ◦ Π, there exists a unique θ-invariant Bowen’s Gibbs state for the potential
+Φγ0 ◦ Π, which we denote by λφ. There exists C ≥ 1 such that for every m ≥ 1,
+any a ∈ Am and any z ∈ [[a]] we have
+C−1 ≤
+λφ[[a]]
+exp
+�
+−P(Φγ0 ◦ Π)m + �m−1
+k=0 Φγ0 ◦ Π(θkz)
+� ≤ C.
+For the series in the denominator of the fraction, for x ∈ Π[[a]] we have
+m−1
+�
+k=0
+Φγ0 ◦ Π(θkΠ−1(x)) = S�m−1
+k=0 R(τ kx)φ(x) − γ0
+m−1
+�
+k=0
+R(τ kx)
+= S∥a∥φ(x) − γ0∥a∥.
+Substituting this and P(Φγ0 ◦ Π) = 0 into the denominator of the fraction implies
+that λφ is a local Gibbs state for the potential φ.
+□
+Remark 2.8. Under the assumption and notation of Proposition 2.7 and its proof,
+if γ0 = P(φ), λφ is θ-invariant and
+�
+Rd(λφ ◦ Π−1) < ∞, then the measure
+1
+�
+Rd(λφ ◦ Π−1)
+∞
+�
+n=0
+(λφ ◦ Π−1)|{R>n} ◦ σ−n
+is in Mφ(X, σ), and it is an equilibrium state for the potential φ. The normalized
+restriction of this measure to Σ is λφ ◦ Π−1.
+
+16
+HIROKI TAKAHASI
+3. Proofs of the main results
+In this section we complete the proofs of all the theorems. In Sections 3.1 and
+3.2, we prove lower and upper bounds for certain fundamental open and closed
+subsets of M(X) respectively. In Section 3.3, we combine these bounds and the
+exponential tightness verified in Section 2 to complete the proof of Theorem A.
+In Sections 3.4 we complete the proof of Theorem B. In view of applications, in
+Section 3.5 we give a sufficient condition for the vanishing of the pressure of the
+induced potential that is assumed in Theorem A. Using this, we complete the proof
+of Theorem C in Section 3.6.
+3.1. Lower bound for fundamental open sets. We introduce notations in this
+and the next two subsections. Let Cu(X) denote the set of real-valued bounded
+uniformly continuous functions on X. For an integer ℓ ≥ 1 we define
+Cu(X)ℓ = {⃗ϕ = (ϕ1, . . . , ϕℓ): ϕj ∈ Cu(X) for every j ∈ {1, . . . , ℓ}}.
+For ⃗ϕ = (ϕ1, . . . , ϕℓ) ∈ Cu(X)ℓ, ⃗α = (α1, . . . , αℓ) ∈ Rℓ and µ ∈ M(X), the
+expression
+�
+⃗ϕdµ > ⃗α indicates that
+�
+ϕjdµ > αj holds for all j ∈ {1, . . . , ℓ}. The
+meaning of
+�
+⃗ϕdµ ≥ ⃗α is analogous. Put ∥⃗α∥ = max1≤j≤ℓ |αj|. For ε ∈ R we write
+⃗ε = (ε, . . . , ε) ∈ Rℓ. For n ≥ 1 and p1 · · · pn ∈ Nn, let p1 · · · pn denote the element
+of En that is contained in [p1 · · · pn].
+Proposition 3.1. Let φ: X → R be acceptable and satisfy P(φ) < ∞. Let ℓ ≥ 1,
+⃗ϕ ∈ Cu(X)ℓ and ⃗α ∈ Rℓ. Let G ⊂ M(X) be an open set of the form
+G =
+�
+µ ∈ M(X):
+�
+⃗ϕdµ > ⃗α
+�
+.
+For any measure µ ∈ Mφ(X, σ) ∩ G, we have
+lim inf
+n→∞
+1
+n log ˜µn(G) ≥ Fφ(µ).
+Proof. By virtue of the definition of the pressure P(φ), it suffices to show that
+(3.1)
+lim inf
+n→∞
+1
+n log
+�
+x∈En
+δn
+x ∈G
+exp Snφ(x) ≥ h(µ) +
+�
+φdµ.
+The proof of [36, Main Theorem] works verbatim to show the next lemma that
+approximates non-ergodic measures with ergodic ones in a particular sense.
+Lemma 3.2. For any µ ∈ Mφ(X, σ) and any ε > 0 there exists an ergodic measure
+µ′ ∈ Mφ(X, σ) which is supported on a compact set and satisfies
+|h(µ) − h(µ′)| < ε,
+����
+�
+⃗ϕdµ −
+�
+⃗ϕdµ′
+���� < ε and
+����
+�
+φdµ −
+�
+φdµ′
+���� < ε.
+By Lemma 3.2, it suffices to show (3.1) for all µ ∈ Mφ(X, σ) which is ergodic. Let
+ε > 0 be such that
+(3.2)
+�
+⃗ϕdµ > ⃗α + ⃗ε.
+
+LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES
+17
+From the uniform continuity of each component of ⃗ϕ and that of φ, from Birkhoff’s
+ergodic theorem and Shannon-McMillan-Breiman’s theorem, for any sufficiently
+large n ≥ 1 there is a finite subset Gn of Nn such that
+(3.3)
+����
+1
+n log #Gn − h(µ)
+���� < ε
+2,
+and for every p1 · · · pn ∈ Gn,
+(3.4)
+sup
+x∈[p1···pn]
+����
+�
+⃗ϕdδn
+x −
+�
+⃗ϕdµ
+���� < ε
+2 and
+sup
+x∈[p1···pn]
+����
+1
+nSnφ(x) −
+�
+φdµ
+���� < ε
+2.
+Then (3.2) and the first inequality in (3.4) yield
+�
+⃗ϕdδn
+p1···pn > ⃗α, and the second
+inequality in (3.4) yields (1/n)Snφ(p1 · · · pn) >
+�
+φdµ − ε/2. Therefore
+�
+x∈En
+δn
+x ∈G
+exp Snφ(x) ≥
+�
+p1···pn∈Gn
+exp Snφ(p1 · · · pn) ≥ #Gn exp
+�
+n
+�
+φdµ − εn
+2
+�
+.
+Taking logarithms and dividing by a sufficiently large n we have
+1
+n log
+�
+x∈En
+δn
+x ∈G
+exp Snφ(x) ≥ 1
+n log #Gn +
+�
+φdµ − ε
+2 > h(µ) +
+�
+φdµ − ε.
+Letting n → ∞ and then ε → 0 yields (3.1).
+□
+3.2. Upper bound for fundamental closed sets. We proceed to upper bounds
+on fundamental closed sets, which are not necessarily compact.
+Proposition 3.3. Let φ: X → R be acceptable and satisfy P(φ) < ∞. Let ℓ ≥ 1,
+⃗ϕ ∈ Cu(X)ℓ, ⃗α ∈ Rℓ and let C ⊂ M(X) be a non-empty closed set of the form
+C =
+�
+µ ∈ M(X):
+�
+⃗ϕdµ ≥ ⃗α
+�
+.
+For any ε > 0 there exists µ ∈ Mφ(X, σ) such that
+�
+⃗ϕdµ > ⃗α − ⃗ε and
+lim sup
+n→∞
+1
+n log ˜µn(C) ≤ Fφ(µ).
+Proof. A main ingredient is the next lemma, the proof of which is analogous to the
+standard proof of the variational principle [40]. For n ≥ 1 we put
+Dn(φ) =
+sup
+p1···pn∈Nn
+sup
+x,y∈[p1···pn]
+Snφ(x) − Snφ(y).
+Lemma 3.4. For any ε > 0 there exists n0 ≥ 1 such that if n ≥ n0 then for any
+non-empty finite subset Cn of Nn satisfying δn
+p1···pn ∈ C for every p1 · · · pn ∈ Cn,
+there exists a measure µ0 ∈ Mφ(X, σ) such that
+log
+�
+p1···pn∈Cn
+sup
+[p1···pn]
+exp Snφ ≤
+�
+h(µ0) +
+�
+φdµ0
+�
+n+Dn(φ)
+and
+�
+⃗ϕdµ0 > ⃗α−⃗ε.
+
+18
+HIROKI TAKAHASI
+Proof. Since all components of ⃗ϕ are bounded uniformly continuous, for any ε > 0
+there exists n0 ≥ 1 such that if n ≥ n0 then for any p1 · · · pn ∈ Nn satisfying
+δn
+p1···pn ∈ C,
+�
+⃗ϕdδn
+x ≥ ⃗α − (1/2)⃗ε holds for any x ∈ [p1 · · · pn]. In what follows we
+assume n ≥ n0.
+Set Λ = �∞
+k=0 σ−nk(�
+p1···pn∈Cn[p1 · · · pn]). Then σn|Λ : Λ → Λ is topologically
+conjugate to the left shift acting on the finite full shift space
+CN
+n = {(ˆpm)∞
+m=1 : ˆpm ∈ Cn for every m ≥ 1}.
+Since the function ˆφ = Snφ induces a continuous potential on CN
+n , the variational
+principle [4] yields
+sup
+ˆµ∈M(Λ,σn|Λ)
+�
+h(ˆµ) +
+�
+ˆφdˆµ
+�
+= lim
+m→∞
+1
+m log
+�
+ˆp1···ˆpm∈Cm
+n
+sup
+[ˆp1···ˆpm]
+�
+exp
+m−1
+�
+k=0
+ˆφ ◦ σnk
+�
+,
+where M(Λ, σn|Λ) denotes the space of σn|Λ-invariant Borel probability measures
+endowed with the weak* topology, and h(ˆµ) denotes the measure-theoretic entropy
+of ˆµ ∈ M(Λ, σn|Λ) with respect to σn|Λ. For the series in the right-hand side, we
+have
+�
+ˆp1···ˆpm∈Cm
+n
+sup
+[ˆp1···ˆpm]
+exp
+�m−1
+�
+k=0
+ˆφ ◦ σnk
+�
+≥
+�
+�
+p1···pn∈Cn
+inf
+[p1···pn] exp Snφ
+�m
+≥
+�
+exp(−Dn(φ))
+�
+p1···pn∈Cn
+sup
+[p1···pn]
+exp Snφ
+�m
+.
+Taking logarithms of both sides, dividing by m and then letting m → ∞ gives
+lim
+m→∞
+1
+m log
+�
+ˆp1···ˆpm∈Cm
+n
+sup
+[ˆp1···ˆpm]
+exp
+�m−1
+�
+k=0
+ˆφ ◦ σnk
+�
+≥ log
+�
+p1···pn∈Cn
+sup
+[p1···pn]
+exp Snφ−Dn(φ).
+Plugging this into the previous inequality yields
+sup
+ˆµ∈M(Λ,σn|Λ)
+�
+h(ˆµ) +
+�
+ˆφdˆµ
+�
+≥ log
+�
+p1···pn∈Cn
+sup
+[p1···pn]
+exp Snφ − Dn(φ).
+By the compactness of the space M(Λ, σn|Λ) and the upper semicontinuity of the
+map ˆµ �→ h(ˆµ) +
+� ˆφdˆµ on this space, the supremum is attained, say by ˆµ0. The
+measure µ0 = (1/n) �n−1
+j=0 ˆµ0 ◦ σ−j is in Mφ(X, σ) and satisfies
+�
+h(µ0) +
+�
+φdµ0
+�
+n =
+sup
+ˆµ∈M(Λ,σn|Λ)
+�
+h(ˆµ) +
+�
+ˆφdˆµ
+�
+.
+Since the support of µ0 is contained in set {x ∈ X :
+�
+⃗ϕdδn
+x > ⃗α − ⃗ε/2} by the
+choice of n0 and the assumption n ≥ n0, we obtain
+�
+⃗ϕdµ0 > ⃗α−⃗ε as required.
+□
+Continuing the proof of Proposition 3.3, we note that P(φ) < ∞ implies Zn(φ) <
+∞ for every n ≥ 1. Hence it is possible to choose a finite subset Cn of the countable
+
+LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES
+19
+set
+�
+p1 · · · pn ∈ Nn : δn
+p1···pn ∈ C
+�
+such that
+�
+p1···pn∈Nn
+δn
+p1···pn∈C
+exp Snφ(p1 · · · pn) ≤ 2
+�
+p1···pn∈Cn
+exp Snφ(p1 · · · pn).
+By this inequality and Lemma 3.4, there exists µ0 ∈ Mφ(X, σ) such that
+�
+⃗ϕdµ0 >
+⃗α − ⃗ε and
+log
+�
+x∈En
+δn
+x ∈C
+exp Snφ(x) = log
+�
+p1···pn∈Nn
+δn
+p1···pn∈C
+exp Snφ(p1 · · · pn)
+≤ log
+�
+p1···pn∈Cn
+exp Snφ(p1 · · · pn) + log 2
+≤ log
+�
+p1···pn∈Cn
+sup
+[p1···pn]
+exp Snφ + log 2
+≤
+�
+h(µ0) +
+�
+φdµ0
+�
+n + Dn(φ) + log 2.
+Since φ is acceptable, it is uniformly continuous and so Dn(φ) = o(n) (n → ∞).
+Dividing both sides of the above last displayed inequality by n, letting n → ∞
+and combining the result with P(φ) = limn→∞(1/n) log Zn(φ) yields the desired
+inequality.
+□
+3.3. Proof of Theorem A. Let φ: X → R be acceptable and satisfy P(φ) < ∞.
+Assume there exists an induced system for which the induced potentials Φγ, γ ∈ R
+associated with φ are locally H¨older continuous, and there exists γ0 ∈ R such that
+P(Φγ0) = 0.
+Let G be a non-empty open subset of M(X). Since subsets of M(X) of the
+form
+�
+µ ∈ M(X):
+�
+⃗ϕdµ > ⃗α
+�
+with ℓ ≥ 1, ⃗ϕ ∈ Cu(X)ℓ, ⃗α ∈ Rℓ constitute a base
+of the weak* topology of M(X), G is written as the union G = �
+λ Gλ of sets Gλ of
+this form. For each Gλ, Proposition 3.1 gives
+lim inf
+n→∞
+1
+n log ˜µn(Gλ) ≥ sup
+Gλ
+Fφ,
+and hence
+lim inf
+n→∞
+1
+n log ˜µn(G) ≥ sup
+λ
+sup
+Gλ
+Fφ = sup
+G
+Fφ = − inf
+G Iφ,
+as required in (1.1).
+Let C be a compact closed subset of M(X). Let G be an arbitrary open set
+containing C. Since M(X) is metrizable by the bounded Lipschitz metric and C is
+compact, we can choose ε > 0 and finitely many closed sets C1, . . . , Cs of the form
+Ck =
+�
+µ ∈ M(X):
+�
+⃗ϕkdµ ≥ ⃗αk
+�
+with ℓk ≥ 1, ⃗ϕk ∈ Cu(X)ℓk, ⃗αk ∈ Rℓk, so that
+C ⊂ �s
+k=1 Ck ⊂ �s
+k=1 Ck(ε) ⊂ G where Ck(ε) = {µ ∈ M(X):
+�
+⃗ϕkdµ > ⃗αk − ⃗ε}.
+By Lemma 3.4 and Fφ ≤ −Iφ, for 1 ≤ k ≤ s we have
+lim sup
+n→∞
+1
+n log ˜µn(Ck) ≤ − inf
+Ck(ε) Iφ + ε.
+
+20
+HIROKI TAKAHASI
+Then we have
+lim sup
+n→∞
+1
+n log ˜µn(C) ≤ max
+1≤k≤s
+�
+− inf
+Ck(ε) Iφ
+�
++ ε ≤ − inf
+G Iφ + ε.
+Since ε > 0 is arbitrary and G is an arbitrary open set containing C, it follows that
+lim sup
+n→∞
+1
+n log ˜µn(C) ≤ inf
+G⊃C
+�
+− inf
+G Iφ
+�
+= − inf
+C Iφ,
+as required in (1.2). The last equality is due to the lower semicontinuity of Iφ.
+Since {˜µn}∞
+n=1 is exponentially tight by Proposition 2.6, the standard arguments
+as in [7] show the upper bound (1.2) for any non-compact closed subset of M(X),
+and that Iφ is a good rate function. This completes the proof of Theorem A.
+□
+3.4. Proof of Theorem B. Let φ: X → R be acceptable and satisfy P(φ) < ∞.
+Assume there exists an induced system for which the associated induced potentials
+Φγ, γ ∈ R are locally H¨older continuous, and there exists γ0 ∈ R such that
+P(Φγ0) = 0. Assume that the minimizer of the rate function Iφ in (1.10) is unique,
+denoted by µmin. Let {˜µnj}∞
+j=1 be an arbitrary convergent subsequence of {˜µn}∞
+n=1
+with the limit measure ˜µ. It suffices to show that ˜µ is the unit point mass at µmin.
+We fix a metric which generates the weak* topology on M(X). Since the rate
+function Iφ in (1.10) is the good rate function by Theorem A, for any α > 0 the
+level set
+L(α) = {µ ∈ M(X): Iφ(µ) ≤ α}
+is a compact set. Let µ ∈ M(X) \ {µmin}. By the lower semicontinuity of the
+rate function and Iφ(µ) > 0, it is possible to take r > 0 such that the closure of
+the open ball Br(µ) of radius r about µ does not intersect L(Iφ(µ)/2). The weak*
+convergence ˜µnj → ˜µ gives
+˜µ(Br(µ)) ≤ lim inf
+j→∞ ˜µnj(Br(µ)).
+By this and the large deviations upper bound for closed sets (1.2), we have
+˜µ(Br(µ)) ≤ lim sup
+j→∞
+˜µnj(Br(µ)) ≤ lim sup
+j→∞
+exp
+�
+−Iφ(µ)nj
+2
+�
+= 0.
+Hence, the support of ˜µ does not contain µ. Since µ is an arbitrary element of
+M(X) \ {µmin}, it follows that ˜µ is the unit point mass at µmin. This completes
+the proof of Theorem B.
+□
+3.5. Sufficient condition for vanishing of pressure. A direct check of the
+condition P(Φγ0) = 0 in Theorem A may be cumbersome, while checking the
+finiteness of induced pressures is considered to be easier. In view of applications,
+we give a sufficient condition for the second assumption in Theorem A on the
+induced potential.
+Lemma 3.5. Let φ: X → R be acceptable and satisfy P(φ) < ∞. Let (Σ, τ|Σ) be
+an induced system and let Φγ : Σ → R (γ ∈ R) be the associated family of induced
+potentials. If there exists δ ∈ R such that 0 < P(Φδ) < ∞, then there exists γ0 ∈ R
+such that P(Φγ0) = 0.
+
+LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES
+21
+Proof. Let γ ∈ R and suppose P(Φγ) < ∞. Let n ≥ 1. Since ∥a∥ ≥ n for any
+a ∈ An and P(Φγ) is finite, for any γ′ > γ we have
+�
+a∈An
+sup
+[a]
+exp(S∥a∥φ − γ′∥a∥) ≤ exp(−(γ′ − γ)n)
+�
+a∈An
+sup
+[a]
+exp(S∥a∥φ − γ∥a∥).
+Taking logarithms, dividing by n and letting n → ∞ yields P(Φγ′) ≤ −γ′ +
+γ + P(Φγ).
+This and the assumption in the lemma together imply that both
+γ∞ = inf{γ ∈ R: P(Φγ) < ∞} and γ0 = inf{γ > γ∞: P(Φγ) ≥ 0} are finite. By
+the variational principle [18, Theorem 2.1.8], γ ∈ (γ∞, ∞) �→ P(Φγ) is convex and
+so continuous. Hence P(Φγ0) = 0 holds.
+□
+3.6. Proof of Theorem C. Recall that T : [0, 1) → [0, 1) denotes the R´enyi map
+(1.11).
+For a bounded interval J ⊂ R let |J| denote its Euclidean length.
+In
+consideration of the neutral fixed point 0 of T, we set N∗ = N \ {1}, N∗ = {1} and
+define an inducing scheme (X∗, R) by (1.5), (1.6), the induced system (Σ, τ|Σ) by
+(1.7), (1.8), and define an infinite alphabet A and a coding map Π by (2.1), (2.2)
+respectively, keeping the notation in Section 2.1. We have Σ = (1/2, 1)∩I and A =
+��∞
+q=2[p1n−1q]: p ≥ 2 and n ≥ 1
+�
+.
+For simplicity we will denote by C various
+positive constants which depend only on T.
+For each a = �∞
+q=2[p1n−1q] ∈ A, we put
+J(a) = T −1([1/(∥a∥ + 1), 1/∥a∥)) ∩ Jp.
+Then R equals ∥a∥ on the set Π[[a]] = π−1(J(a)). There exists C ≥ 1 such that for
+any a ∈ W(A) we have
+(3.5)
+C−1 ≤ |J(a)| · ∥a∥2 ≤ C.
+Define an induced map U : �
+a∈A J(a) → [0, 1) by U|J(a) = T ∥a∥|J(a). Recall that
+φ = − log |T ′ ◦ π|. For β, γ ∈ R define Φβ,γ : Σ → R by
+Φβ,γ(x) = βSR(x)φ(x) − γR(x),
+which is the induced potential associated with βφ.
+Lemma 3.6. For all β, γ ∈ R, Φβ,γ is locally H¨older continuous.
+Proof. From the bounded distortion near the neutral fixed point [19, Lemma 2.2],
+there exists C > 0 such that for any a ∈ A and all x, y ∈ [a] we have
+(3.6)
+Φβ,γ(x) − Φβ,γ(y) ≤ Cβ|U(ξ) − U(η)| ≤ Cβ,
+where ξ = π(x) and η = π(y). If x ̸= y then d(x, y) = e−n holds for some n ≥ 2,
+and there exists a1 · · ·an ∈ An such that x, y ∈ [[a1 · · · an]]. Since there is ρ > 1
+such that inf[0,1)\J1 |T ′| ≥ ρ, if n ≥ 3 then we have
+(3.7)
+|U(ξ) − U(η)| ≤
+|Un−1(ξ) − Un−1(η)|
+inf�n−1
+k=2 U−k(J(ak)) |(Un−2)′| ≤ ρ2−n.
+The local H¨older continuity of Φβ,γ follows from (3.6) and (3.7).
+□
+Lemma 3.7. For any β ∈ (1/2, 1] there exists γ ∈ R such that 0 ≤ P(Φβ,γ) < ∞.
+
+22
+HIROKI TAKAHASI
+Proof. From Lemma 3.6, |T(Jp)| = 1 for p ≥ 2 and (3.5), there exists C ≥ 1 such
+that for a = �∞
+q=2[p1n−1q] ∈ A and all β, γ ∈ R we have
+1
+|Jp|β sup
+[a]
+exp Φβ,γ = e−γn sup
+Π[a]
+exp(βSnφ)
+|Jp|β
+≤ Ce−γnn−2β.
+Summing the result over all a ∈ A, we have
+∞
+�
+n=1
+�
+a∈A
+∥a∥=n
+sup
+[a]
+exp Φβ,γ ≤ C
+∞
+�
+n=1
+e−γnn−2β
+∞
+�
+p=2
+|Jp|β ≤ C
+∞
+�
+n=1
+e−γnn−2β
+∞
+�
+p=2
+p−2β.
+Let β ∈ (1/2, 1). Then we have P(βφ) > 0 [13], and the above series is finite
+for all γ ∈ (0, P(βφ)]. In particular, P(Φβ,P (βφ)) is finite. Since any measure in
+M(X, σ) other than the unit point mass at 1∞ = 111 · · · charges Σ, the equilibrium
+state µβφ for the potential βφ satisfies µβφ(Σ) > 0. Let ˆµβφ denote the normalized
+restriction of µβφ to Σ. Since τ is the first return map to Σ, ˆµβφ is τ|Σ-invariant
+and satisfies
+�
+Rdˆµβφ < ∞. By the variational principle for Φβ,P (βφ) and Abramov-
+Kac’s formula [44, Theorem 5.1], we obtain
+∞ > P(Φβ,P (βφ)) ≥ h(ˆµβφ) +
+�
+(Φβ − P(βφ)R)dˆµβφ
+= (Fβφ(µβφ) + P(βφ) − P(βφ))
+�
+Rdˆµβφ = 0.
+We have verified1 that 0 ≤ P(Φβ,P (βφ)) < ∞ as reqiured in the lemma.
+For the remaining case β = 1, we have P(φ) = 0 [13]. From Lemma 3.6 there is
+C ≥ 1 such that for n ≥ 1 and a = a1 · · · an ∈ An,
+C−1 ≤
+���n
+k=1 U−k(J(ak))
+��
+sup[a] exp
+��n−1
+k=0 Φ1,0 ◦ τ k� ≤ C.
+Summing this double inequalities over all a ∈ An, taking logarithms, dividing by
+n and then letting n → ∞ yields P(Φ1,0) = 0 as required in the lemma.
+□
+Lemma 3.6 and Lemmas 3.5, 3.7 together verify the assumption in Theorem A
+for the potential βφ, β ∈ (1/2, 1]. It follows from [35] that for any β ∈ (1/2, 1],
+any minimizer of the rate function Iβφ is an equilibrium state for βφ. Since the
+equilibrium state for βφ is unique [13], the minimizer of the rate function Iβφ is
+unique. Since the map π in (1.13) is continuous, the assertions in Theorem C
+follow from Theorems A and B.
+□
+3.7. Some generalizations. We have worked on two full shift spaces X and Σ
+(or AN), the latter obtained from the former via inducing (recall Section 2.1). The
+assumption that X is the full shift has been used to construct sets of periodic
+points of the same period, in the proofs of exponential tightness (Lemma 2.5) and
+the lower large deviation bound (Proposition 3.1). For the induced system (Σ, τ|Σ),
+we have effected the thermodynamic formalism for countable Markov shifts [18].
+1In fact, one can show P(Φβ,P (βφ)) = 0. See [23] for example.
+
+LEVEL-2 LDP FOR COUNTABLE MARKOV SHIFTS WITHOUT GIBBS STATES
+23
+The setup in this paper can be slightly generalized. The above-mentioned con-
+structions of sets of periodic points can be done even if X is replaced by a finitely
+primitive shift (see [18] for the definition). Then the induced shift space becomes
+finitely irreducible, for which the thermodynamic formalism works too [27].
+Acknowledgments. This research was partially supported by the JSPS KAK-
+ENHI 19K21835 and 20H01811.
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+Email address: hiroki@math.keio.ac.jp
+

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+MNRAS 000, 1–5 (2023)
+Preprint 5 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Dust processing in protoplanetary envelopes as the origin of hot
+minerals in comets
+Mohamad Ali-Dib1⋆
+1Center for Astro, Particle and Planetary Physics (CAP3), New York University Abu Dhabi, UAE
+Accepted 2022-12-30
+ABSTRACT
+Crystalline silicates are found in a large number of comets. These pose a long-standing
+conundrum for solar system formation models as they can only be created in the in-
+ner hot disk at temperatures higher than 800 K, and there is no obvious mechanism
+to transport them out into the comets formation region. Here we propose that these
+particles could have formed inside the hydrostatic envelopes surrounding young pro-
+toplanets still embedded in the protoplanetary disk. Using a simplified 1D model we
+investigate the thermal structure of these envelopes, and find that for core masses
+ranging from 0.08 to 1.5 M⊕, located anywhere between 1 and 30 AU, the temper-
+ature and pressure at the base of the envelopes are high enough to quickly vaporize
+silicate particles of various sizes. Moreover, if the grain abundance is atleast solar, these
+envelopes become fully convective, allowing for dust ejection across the Bondi radius
+back into the disk. Amorphous silicates are hence thermally processed into crystalline
+particles in these envelopes, and then transported back to disk through convective
+diffusion to be finally incorporated into the cometary building blocks.
+Key words: planets and satellites: formation – comets: general – planets and satellites:
+composition
+1
+INTRODUCTION
+High temperature minerals are ubiquitous in the cold outer
+solar system small bodies. One of the earliest remote sensing
+detections was for crystalline silicates (CSs) such as olivines
+and pyroxenes in the grains of comets 1P/Halley, D/1993
+F2 (Shoemaker-Levy), C/1987 P1 (Bradfield) (Hanner et al.
+1994), and C/1993 A1 (Mueller) (Hanner, Lynch, & Russell
+1994). Subsequent detections were made in comets C/1995
+O1 (Hale-Bopp) (Hayward, Hanner, & Sekanina 2000; Cro-
+visier et al. 1997; Wooden et al. 1999), 103P/Hartley (Cro-
+visier et al. 2000), and more recently 17P/Holmes (Shinnaka
+et al. 2018). On the other hand Calcium-Aluminum inclu-
+sions (CAIs) that form at even higher temperatures were
+found in the dust collected by Stardust in comet 81P/Wild
+(Brownlee et al. 2006). We refer the reader to the observa-
+tional review of Mumma & Charnley (2011) for more infor-
+mations.
+The presence of CSs have been a primary challenge to
+solar system formation models for decades, as the thermal
+conditions in the outer protoplanetary disk are not con-
+ducive to their formation locally. CSs can form starting
+from amorphous silicates through either direct vaporization
+followed by re-condensation at temperatures higher than
+⋆ E-mail: malidib@nyu.edu
+∼1800 K, or thermal annealing for T> 800 K. Annealing is
+a physical process where sufficiently energetic molecules of a
+solid slowly regroup into a crystal lattice. This mechanism is
+not instantaneous and necessitates high temperature expo-
+sure for a period of few weeks followed by slow cooling (Gail
+1998, 2001). In typical protoplanetary disks, direct conden-
+sation can be active only inside ∼ 0.1 AU, and annealing is
+inefficient outside ∼ 1.5 AU.
+Moreover, there is no acceptable mechanism to trans-
+port these particles from the inner disk to the comets for-
+mation region. Earlier transport models relied on turbulent
+diffusion (Bockel´ee-Morvan et al. 2002), but recent ALMA
+observations suggest that protoplanetary disks are laminar
+(Flaherty et al. 2015, 2018), and thus this mechanism is un-
+likely to be efficient. Large scale outward advection in the
+disk’s midplane has also been proposed (Hughes & Armitage
+2010), but 3D MHD simulations ruled out the presence of
+such advection (Fromang, Lyra, & Masset 2011). Another
+possible transport mechanism is photophoresis (Mousis et
+al. 2007), but this necessitates a relatively large (1-2 AU)
+central hole in the disk. Finally, Ali-Dib et al. (2015) pro-
+posed that FU-Ori outbursts might form these particles in
+situ, but this depends on the outburst trigger radius being
+large enough, which is uncertain.
+Here we show how high temperature minerals form nat-
+urally, and in-situ, in the envelopes surrounding low mass
+© 2023 The Authors
+arXiv:2301.01472v1  [astro-ph.EP]  4 Jan 2023
+
+2
+M. Ali-Dib
+proto-planets embedded in the disk. We present models
+showing that the temperatures and pressures at the base
+of these envelopes easily reach conditions that allow for
+the formation of crystalline silicates through direct vapor-
+ization and re-condensation. Primordial amorphous silicates
+are thus accreted and then thermally processed in these en-
+velopes, before finally getting ejected back to the disk as
+crystalline particles via convective diffusion. We emphasize
+that this work concerns the formation of generic CSs such as
+olivines and pyroxenes, and not necessarily chondrules and
+CAIs due to their additional formation-time constrains that
+are outside the scope of this work. We present our model in
+section 2, results in section 3, and conclude in section 4.
+2
+MODEL
+We model the atmosphere using the standard atmospheric
+structure equations. The equation of hydrostatic equilibrium
+is given by:
+dP
+dr = −GM
+r2 ρ(r)
+(1)
+and we define the temperature gradient equation starting
+from the standard assumption that heat can be transported
+using either radiation (if the local envelope is convectively
+stable) or adiabatic convection (if unstable). It is hence writ-
+ten as:
+dT
+dr = ∇ T
+P
+dP
+dr
+(2)
+where ∇ is defined, starting from the Schwarzschild con-
+vective stability criterion (∇rad
+<
+∇ad), to be ∇
+=
+min(∇ad, ∇rad). Here ∇ad is the adiabatic gradient:
+∇ad ≡
+� d ln T
+d ln P
+�
+ad
+= γ − 1
+γ
+(3)
+where the adiabatic constant γ = 1.5. ∇rad is the radiative
+gradient:
+∇rad ≡
+3κP
+64πGMσT 4 L
+(4)
+where L is the envelope’s luminosity generated by accretion
+at a rate
+˙Macc :
+L = GM ˙Macc
+Rc
+(5)
+Rc is the core radius, and κ is the opacity that we define
+following Ormel (2014) as:
+κ = κgas + κgr
+(6)
+with:
+κgr = κgeomQe = 3Zgr
+4ρsa × min(0.6πa
+λmax , 2)
+(7)
+where Zgr is the grains abundance, as their size, and ρs their
+internal density. we use ρs = 3 g/cm3 for both the core and
+the dust particles.
+The equilibrium dust size in the envelope as is set by
+two competing processes: grain growth through coagulation
+(Ormel 2014) and grain collisional destruction (Ali-Dib &
+Thompson 2020). The relative relevance of these two pro-
+cesses is decided mainly by whether the collisional speeds
+reach the silicate fragmentation threshold (Vf ∼ 100 cm/s,
+Blum & Wurm (2008)). The collisional speed is approxi-
+mated here as the largest among the dust’s convective ve-
+locity Vcon,d (eq. 19) and the dust’s radial drift velocity:
+Vdrift,d = τstop GM
+r2
+(8)
+where τstop is the stopping time.
+As discussed in (Ali-Dib & Thompson 2020), collisions
+in these envelopes are likely to be destructive. This leads
+to a small characteristic dust size, increasing the opac-
+ity (thus growing the convective zone), and decreasing the
+vaporization timescale. Here we only select models where
+max(Vcon,d,Vdrift,d) is higher than 100 cm/s everywhere in
+the disk.
+The convective fragmentation dust size is hence calcu-
+lated following (Ali-Dib & Thompson 2020) as:
+as,conv = 4πV 2
+f r3ρ2
+gcg
+Lρs
+(9)
+where ρg and cg are the gas’ density and sound speed. The
+drift fragmentation dust size is given by:
+as,drift = Vfr2ρgcg
+GMρs
+(10)
+with finally as=min(as,conv,as,drift).
+Note that as,conv is defined everywhere in the envelope,
+since, as discussed below, we also only select fully convec-
+tive envelopes. For this approach to be applicable, the par-
+ticles need to reach the local fragmentation threshold at ev-
+ery point in the envelope. Therefore, for self-consistency, we
+only keep models where the mean free time for collisions
+is shorter than the convective timescale. In the convective
+fragmentation regime this can be written as:
+as
+as + 4ℓg/9 < 9Z2
+gr
+ρg
+ρs Mcon r
+ℓg
+(11)
+where Mcon is the convective Mach number, and ℓg the mean
+free path of the gas. In the drift regime this is replaced by :
+3
+4Zgr
+c2
+gr
+GM
+Mcon
+1 + 9as/4ℓg < 1
+(12)
+The gas opacity is given by:
+κgas = 10−8ρ2/3
+g
+T 3
+(13)
+Finally we close the system with the ideal gas equation of
+state P = ρgkBT/µ. We solve these equations by integrating
+inwards from the outer boundary at Rout, the minimum of
+the Bondi and Hill radii, to the core. We assumed the disk is
+radiative and calculate its temperature and density following
+Ali-Dib, Cumming, & Lin (2020):
+Td = 373 r−9/10
+au
+K and ρd = 1.7×10−10 r−33/20
+au
+g/cm3 (14)
+Once we have the envelope’s thermal structure, we can
+calculate additional quantities needed for the subsequent
+analysis. We calculate the silicate particles vaporization rate
+as :
+1
+as
+das
+dt = −
+� µSil
+2πkT
+�1/2 P sat
+Sil
+ρsas
+(15)
+MNRAS 000, 1–5 (2023)
+
+Origins of hot minerals
+3
+with (Krieger 1967):
+P sat
+sil (T) = 3.2 × 1014e−(6×104 K)/T
+(16)
+and hence the silicate grains vaporization timescale is given
+by :
+τvap,sil =
+� 1
+as
+das
+dt
+�−1
+(17)
+We define the gas and dust convective velocities respec-
+tively as:
+Vcon,g =
+�
+L
+4πr2ρg
+�1/3
+(18)
+where we assumed that in the convective zone the energy is
+entirely transported through adiabatic convection, and
+Vcon,d ∼ Vcon (Vconτstop/r)1/2
+(19)
+We
+finally
+calculate
+the
+dust’s
+convective
+mixing
+timescale as:
+τmix,d = H/Vcon,d
+(20)
+3
+RESULTS
+We start by exploring parameter space in order to find the
+values that allow for the creation of CSs in proto-envelopes.
+We explore core masses ranging from Pluto’s mass (0.002
+M⊕) to a hypothetical giant planet’s core (10 M⊕), placed
+between 1 and 30 AU where ambient temperatures are too
+low to create CSs in the disk. The grains abundance Zgr
+ranges from subsolar (10−3) to supersolar (1.0).
+Our results are summarized in Fig. 1. In this plot we
+show only the areas of parameter space leading to envelopes
+conducive to the creation of CSs and that are self-consistent
+to our model assumptions. This is defined by these condi-
+tions:
+(i) τvap,sil is less than τmix,d at the base of the envelope.
+This simply constrain the envelopes to those where solid
+silicates at their base can get vaporized faster than they are
+transported back into the upper cooler zones.
+(ii) The envelope is fully convective. This ensures that the
+newly created CSs can be convectively diffused all the way
+back into the disk. This condition is inspired by the results of
+Ali-Dib & Thompson (2020) who considered a similar setup
+with a 0.3 M⊕ core embedded in the disk, and showed that,
+for typical accretion rates, pebble fragmentation and dust
+loading increases the opacity and push the convective zone
+out till it reaches the Bondi radius. Dust particles in these
+steady-state envelopes are then diffusively ejected back to
+the disk. Our results rely on this mechanism to transport
+the newly created CSs from the hot inner envelope back to
+the disk to be incorporated in proto-comets.
+(iii) The collisional velocity is higher than 100 cm/s
+throughout the envelope, and conditions 11 and 12 are sat-
+isfied. This ensures that our dust size prescription is self-
+consistent.
+3.1
+Core mass
+Figure 1 shows that, while CSs can be created under a va-
+riety of parameter ranges, trends do exist. We start with
+our nominal model, for
+˙Macc = 10−6M⊕/yr. First, there
+is a relatively narrow range of masses that extends from
+around 0.08 M⊕ (40 times Pluto’s mass) to 1.5 M⊕, beyond
+which the chances of creating CSs drops drastically. This
+implies that CSs might have formed in the proto-envelopes
+of Mars to Earth mass protoplanets that have since disap-
+peared via giant collisions or dynamical ejection, or possibly
+grown into giant planetary cores. The lower limit on core
+masses is mainly due to their envelopes’ relatively cooler
+temperatures, increasing τvap,sil considerably. On the other
+hand, cores with masses higher than 1.5 M⊕ have dust par-
+ticles large enough in their middle and inner envelopes to
+switch from the Rosseland mean opacity regime to the ge-
+ometric opacity regime, as can be seen in Fig. 2 (left hand
+panel). This decreases the radiative gradient, creating an in-
+ner radiative zone that prevents these envelopes from being
+fully convective. It is worth noting that, while our model
+considers the smallest of the Hill and Bondi radii to be the
+envelope’s outer boundary, all of our acceptable cases that
+form CSs are in the Bondi regime. This is expected as the
+Hill regime dominates for higher mass cores (5-10 M⊕) that
+were excluded above. The Bondi radius RB = 2GM/c2
+s is
+obtained by equating the local sound speed to the gravi-
+tational escape velocity, and thus describes a usually light
+but bound envelope where gas particles do not have enough
+thermal energy to escape. For higher mass cores, the Bondi
+radius is large enough for the Hill stability criteria to become
+the more stringent constrain.
+3.2
+Semimajor axis
+A complementary piece of information is the semimajor axis,
+where we find that CSs can form almost anywhere in the disk
+if the envelope’s grain abundance is high enough as discussed
+further below. Semimajor axis controls the temperature and
+density at the outer boundary, which seems to be important
+only in the marginal cases, for example for low core masses
+where the envelopes would be too cold if placed further out
+in the disk. The wide range of possible semimajor axis allows
+for the possibility of creating CSs in the comets formation
+region. Classically, Oort cloud comets were thought to form
+among the giant planets all the way down to 5 AU, while
+Jupiter family comets were thought to form in the scattered
+disk (Duncan, Quinn, & Tremaine 1987; Duncan & Levison
+1997; Dones et al. 2015). Alternatively, Brasser & Morbidelli
+(2013) proposed that both could have formed in the same
+region beyond Neptune.
+3.3
+Grains abundance
+We moreover find that creating CSs necessitate solar to su-
+persolar grain abundance in the envelope (Zgr >= 0.01).
+This result is not consistent with the subsolar grain abun-
+dances found in models that incorporate dust growth & set-
+tling to the core but omit dust fragmentation with convec-
+tive mixing. Ormel (2014) for example added a simple grain
+growth equation to the atmospheric structure equations, and
+found that Zgr can be as low as 10−4 in parts of the envelope.
+Mordasini (2014) also created an atmospheric model incor-
+porating dust settling and coagulation, and found that this
+mostly results in subsolar opacities. The main role of Zgr is
+MNRAS 000, 1–5 (2023)
+
+4
+M. Ali-Dib
+to increase the opacity and extend the convective zone all the
+way to the outer boundary (Bondi or Hill radius). Ali-Dib
+& Thompson (2020) discussed the gradual buildup of Zgr in
+the envelope through accretion and fragmentation, and de-
+rived a lower limit on Zgr in order to get a fully convective
+envelope:
+Zgr > 0.12 T 2
+d,2
+ρd,−11
+�tacc,c
+Myr
+� �
+Mc
+0.3M⊕
+�−2/3
+(21)
+which is generally consistent with our Zgr values. In order to
+get supersolar Zgr, multiple conditions need to be satisfied:
+• The dust size need to be fragmentation-limited, which
+is a pre-requisite for our dust-size prescription. This depends
+on many factors, including the accretion rate (setting the lu-
+minosity and thus convective speeds) and particles’ porosity
+and chemical composition (Blum & Wurm 2008; Okuzumi
+et al. 2012; Wada et al. 2008, 2009).
+• A significant fraction of the dust should not get accreted
+by the core, but remain mixed in the envelope. This is an
+open question with many complications. In our cases, sili-
+cates are in vapor form at the base of the envelope which
+should stop accretion from taking place unless the tempera-
+ture is low enough for the inner envelope to reach saturation
+pressure. This also depends on the nature of convection in
+these envelopes, whether it is diffusive as we are assuming,
+or whether it is dominated by large scale eddies that can
+enhance accretion by the core (Johansen & Nordlund 2020).
+3.4
+Accretion rate
+Finally we investigate the effects of using a lower accretion
+rate. Our results for
+˙Macc = 10−7M⊕/yr are shown in Fig.
+1. In this case we find that while the semimajor axis range
+remains the same and the lower mass limit does not change
+(∼ 0.08M⊕), the upper limit decreases by over a factor 2 to
+∼ 0.6M⊕. This is expected since, lower accretion rate leads
+to lower luminosities. As seen in Fig. 2 (right hand side),
+this decreases the radiative gradient and allows for a radia-
+tive zone in the inner envelope even though the dust size in
+the 2 cases converge to the same inner value. In some cases
+Zgr can compensate for the lower luminosity and increases
+the opacity enough to create fully convective envelopes, ex-
+plaining the overall larger Zgr we find for the lower accretion
+rate cases.
+4
+SUMMARY & CONCLUSIONS
+Crystalline silicates are ubiquitous in comets, but can only
+form at very high temperatures. Here we investigated the
+possibility of transforming amorphous silicates into crys-
+talline particles inside the envelopes of protoplanets through
+vaporization followed by re-condensation, and then ejecting
+them back to the disk through diffusion in the fully con-
+vective envelopes. Using a simplified 1D envelope structure
+model that incorporates a dust size prescription accounting
+for fragmentation and growth, we showed that crystalline
+silicates can be created from a diverse set of parameters.
+Cores need to be between 0.08 to 1.5 M⊕ in mass, as lighter
+cores do not allow for temperatures high enough to vaporize
+silicates, and the envelopes of more massive cores are often
+not fully convective. We finally found that the location in
+the disk (1 to 30 AU) has little influence on the results, ex-
+cept in marginal cases, and that a solar to supersolar grain
+abundance is needed, but this can be achieved through dust
+fragmentation and accumulation. Our mechanism is simple
+and does not rely on assumptions about the disk, although
+it depends on the assumed diffusive nature of 1D convection.
+Whether this is realistic needs to be investigated further us-
+ing 3D hydrodynamic simulations.
+ACKNOWLEDGEMENTS
+We thank the anonymous referee for their constructive com-
+ments that greatly improved this manuscript. This work is
+supported by Tamkeen under the NYU Abu Dhabi Research
+Institute grant CAP3.
+DATA AVAILABILITY
+The data underlying this article (numerical simulations out-
+put files) will be shared on reasonable request to the corre-
+sponding author.
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+Origins of hot minerals
+5
+Figure 1. The semimajor axis, core mass, and envelope grain abundance for all the cases that satisfy our conditions to form and eject
+crystal silicates as enumerated in section 3. Left:
+˙Macc = 10−6M⊕/yr . Right:
+˙Macc = 10−7M⊕/yr. Note the different color scales for
+the two panels.
+109
+1010
+1011
+r [cm]
+10-4
+10-3
+10-2
+10-1
+100
+101
+102
+103
+104
+105
+2π ad / λmax
+2π ad / λmax,  5M⊕
+2π ad / λmax, 1M⊕
+Geometric opacity limit
+10-1
+100
+101
+102
+Temperature gradient
+∇rad, 1M⊕
+∇rad, 5M⊕
+∇ad
+109
+1010
+1011
+r [cm]
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+Dust size [cm]
+ad, ˙M=10−6
+ad, ˙M=10−7
+10-1
+100
+101
+102
+Temperature gradient
+∇rad, ˙M=10−6
+∇rad, ˙M=10−7
+∇ad
+Figure 2. Left: solid lines are the opacity efficiency factors Qe (eq. 7) for 2 different core masses with all other parameters being equal
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+gradients, indicating the radiative and convective zones. Right: solid lines are the dust size ad for cases with 2 different accretion rates
+but all other parameters being equal (15 AU, 1 M⊕). Dashed lines are the radiative and adiabatic gradients for the same cases. In all
+plots, the x-axis is the radius from the core, extending from the core to the envelope’s outer boundary.
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+MNRAS 000, 1–5 (2023)
+
+Macc = 10-6 Ma /yr
+0.0
+-0.2
+100
+Core Mass [M
+-0.6
+-0.8
+-1.0
+ grai
+-1.2
+6
+10-1
+Lo
+-1.4
+-1.6
+0
+5
+10
+15
+20
+25
+30
+Semimajor axis [AU]Macc = 10-7 Mg / yr
+100
+0.00
+-0.08
+-0.16
+b-
+N
+Core Mass [M]
+-0.24
+abundance
+-0.32
+-0.40
+og grain
+10-1
+-0.48
+-0.56
+-0.64
+-0.72
+0
+5
+10
+15
+20
+25
+30
+Semimajor axis [AU]

8tAzT4oBgHgl3EQfgvwn/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf,len=409
+page_content='MNRAS 000, 1–5 (2023)  Preprint 5 January 2023  Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='0  Dust processing in protoplanetary envelopes as the origin of hot  minerals in comets  Mohamad Ali-Dib1⋆  1Center for Astro, Particle and Planetary Physics (CAP3), New York University Abu Dhabi, UAE  Accepted 2022-12-30  ABSTRACT  Crystalline silicates are found in a large number of comets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' These pose a long-standing  conundrum for solar system formation models as they can only be created in the in-  ner hot disk at temperatures higher than 800 K, and there is no obvious mechanism  to transport them out into the comets formation region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Here we propose that these  particles could have formed inside the hydrostatic envelopes surrounding young pro-  toplanets still embedded in the protoplanetary disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Using a simplified 1D model we  investigate the thermal structure of these envelopes, and find that for core masses  ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='08 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='5 M⊕, located anywhere between 1 and 30 AU, the temper-  ature and pressure at the base of the envelopes are high enough to quickly vaporize  silicate particles of various sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Moreover, if the grain abundance is atleast solar, these  envelopes become fully convective, allowing for dust ejection across the Bondi radius  back into the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Amorphous silicates are hence thermally processed into crystalline  particles in these envelopes, and then transported back to disk through convective  diffusion to be finally incorporated into the cometary building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  Key words: planets and satellites: formation – comets: general – planets and satellites:  composition  1  INTRODUCTION  High temperature minerals are ubiquitous in the cold outer  solar system small bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' One of the earliest remote sensing  detections was for crystalline silicates (CSs) such as olivines  and pyroxenes in the grains of comets 1P/Halley, D/1993  F2 (Shoemaker-Levy), C/1987 P1 (Bradfield) (Hanner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  1994), and C/1993 A1 (Mueller) (Hanner, Lynch, & Russell  1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Subsequent detections were made in comets C/1995  O1 (Hale-Bopp) (Hayward, Hanner, & Sekanina 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Cro-  visier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Wooden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 1999), 103P/Hartley (Cro-  visier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 2000), and more recently 17P/Holmes (Shinnaka  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' On the other hand Calcium-Aluminum inclu-  sions (CAIs) that form at even higher temperatures were  found in the dust collected by Stardust in comet 81P/Wild  (Brownlee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' We refer the reader to the observa-  tional review of Mumma & Charnley (2011) for more infor-  mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  The presence of CSs have been a primary challenge to  solar system formation models for decades, as the thermal  conditions in the outer protoplanetary disk are not con-  ducive to their formation locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' CSs can form starting  from amorphous silicates through either direct vaporization  followed by re-condensation at temperatures higher than  ⋆ E-mail: malidib@nyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='edu  ∼1800 K, or thermal annealing for T> 800 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Annealing is  a physical process where sufficiently energetic molecules of a  solid slowly regroup into a crystal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' This mechanism is  not instantaneous and necessitates high temperature expo-  sure for a period of few weeks followed by slow cooling (Gail  1998, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' In typical protoplanetary disks, direct conden-  sation can be active only inside ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='1 AU, and annealing is  inefficient outside ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='5 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  Moreover, there is no acceptable mechanism to trans-  port these particles from the inner disk to the comets for-  mation region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Earlier transport models relied on turbulent  diffusion (Bockel´ee-Morvan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 2002), but recent ALMA  observations suggest that protoplanetary disks are laminar  (Flaherty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 2015, 2018), and thus this mechanism is un-  likely to be efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Large scale outward advection in the  disk’s midplane has also been proposed (Hughes & Armitage  2010), but 3D MHD simulations ruled out the presence of  such advection (Fromang, Lyra, & Masset 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Another  possible transport mechanism is photophoresis (Mousis et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 2007), but this necessitates a relatively large (1-2 AU)  central hole in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Finally, Ali-Dib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' (2015) pro-  posed that FU-Ori outbursts might form these particles in  situ, but this depends on the outburst trigger radius being  large enough, which is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  Here we show how high temperature minerals form nat-  urally, and in-situ, in the envelopes surrounding low mass  © 2023 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='01472v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='EP]  4 Jan 2023  2  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Ali-Dib  proto-planets embedded in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' We present models  showing that the temperatures and pressures at the base  of these envelopes easily reach conditions that allow for  the formation of crystalline silicates through direct vapor-  ization and re-condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Primordial amorphous silicates  are thus accreted and then thermally processed in these en-  velopes, before finally getting ejected back to the disk as  crystalline particles via convective diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' We emphasize  that this work concerns the formation of generic CSs such as  olivines and pyroxenes, and not necessarily chondrules and  CAIs due to their additional formation-time constrains that  are outside the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' We present our model in  section 2, results in section 3, and conclude in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  2  MODEL  We model the atmosphere using the standard atmospheric  structure equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' The equation of hydrostatic equilibrium  is given by:  dP  dr = −GM  r2 ρ(r)  (1)  and we define the temperature gradient equation starting  from the standard assumption that heat can be transported  using either radiation (if the local envelope is convectively  stable) or adiabatic convection (if unstable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' It is hence writ-  ten as:  dT  dr = ∇ T  P  dP  dr  (2)  where ∇ is defined, starting from the Schwarzschild con-  vective stability criterion (∇rad  <  ∇ad), to be ∇  =  min(∇ad, ∇rad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Here ∇ad is the adiabatic gradient:  ∇ad ≡  � d ln T  d ln P  �  ad  = γ − 1  γ  (3)  where the adiabatic constant γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' ∇rad is the radiative  gradient:  ∇rad ≡  3κP  64πGMσT 4 L  (4)  where L is the envelope’s luminosity generated by accretion  at a rate  ˙Macc :  L = GM ˙Macc  Rc  (5)  Rc is the core radius, and κ is the opacity that we define  following Ormel (2014) as:  κ = κgas + κgr  (6)  with:  κgr = κgeomQe = 3Zgr  4ρsa × min(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='6πa  λmax , 2)  (7)  where Zgr is the grains abundance, as their size, and ρs their  internal density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' we use ρs = 3 g/cm3 for both the core and  the dust particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  The equilibrium dust size in the envelope as is set by  two competing processes: grain growth through coagulation  (Ormel 2014) and grain collisional destruction (Ali-Dib &  Thompson 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' The relative relevance of these two pro-  cesses is decided mainly by whether the collisional speeds  reach the silicate fragmentation threshold (Vf ∼ 100 cm/s,  Blum & Wurm (2008)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' The collisional speed is approxi-  mated here as the largest among the dust’s convective ve-  locity Vcon,d (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 19) and the dust’s radial drift velocity:  Vdrift,d = τstop GM  r2  (8)  where τstop is the stopping time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  As discussed in (Ali-Dib & Thompson 2020), collisions  in these envelopes are likely to be destructive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' This leads  to a small characteristic dust size, increasing the opac-  ity (thus growing the convective zone), and decreasing the  vaporization timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Here we only select models where  max(Vcon,d,Vdrift,d) is higher than 100 cm/s everywhere in  the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  The convective fragmentation dust size is hence calcu-  lated following (Ali-Dib & Thompson 2020) as:  as,conv = 4πV 2  f r3ρ2  gcg  Lρs  (9)  where ρg and cg are the gas’ density and sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' The  drift fragmentation dust size is given by:  as,drift = Vfr2ρgcg  GMρs  (10)  with finally as=min(as,conv,as,drift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  Note that as,conv is defined everywhere in the envelope,  since, as discussed below, we also only select fully convec-  tive envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' For this approach to be applicable, the par-  ticles need to reach the local fragmentation threshold at ev-  ery point in the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Therefore, for self-consistency, we  only keep models where the mean free time for collisions  is shorter than the convective timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' In the convective  fragmentation regime this can be written as:  as  as + 4ℓg/9 < 9Z2  gr  ρg  ρs Mcon r  ℓg  (11)  where Mcon is the convective Mach number, and ℓg the mean  free path of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' In the drift regime this is replaced by :  3  4Zgr  c2  gr  GM  Mcon  1 + 9as/4ℓg < 1  (12)  The gas opacity is given by:  κgas = 10−8ρ2/3  g  T 3  (13)  Finally we close the system with the ideal gas equation of  state P = ρgkBT/µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' We solve these equations by integrating  inwards from the outer boundary at Rout, the minimum of  the Bondi and Hill radii, to the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' We assumed the disk is  radiative and calculate its temperature and density following  Ali-Dib, Cumming, & Lin (2020):  Td = 373 r−9/10  au  K and ρd = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='7×10−10 r−33/20  au  g/cm3 (14)  Once we have the envelope’s thermal structure, we can  calculate additional quantities needed for the subsequent  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' We calculate the silicate particles vaporization rate  as :  1  as  das  dt = −  � µSil  2πkT  �1/2 P sat  Sil  ρsas  (15)  MNRAS 000, 1–5 (2023)  Origins of hot minerals  3  with (Krieger 1967):  P sat  sil (T) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='2 × 1014e−(6×104 K)/T  (16)  and hence the silicate grains vaporization timescale is given  by :  τvap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='sil =  � 1  as  das  dt  �−1  (17)  We define the gas and dust convective velocities respec-  tively as:  Vcon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='g =  �  L  4πr2ρg  �1/3  (18)  where we assumed that in the convective zone the energy is  entirely transported through adiabatic convection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' and  Vcon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='d ∼ Vcon (Vconτstop/r)1/2  (19)  We  finally  calculate  the  dust’s  convective  mixing  timescale as:  τmix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='d = H/Vcon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='d  (20)  3  RESULTS  We start by exploring parameter space in order to find the  values that allow for the creation of CSs in proto-envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  We explore core masses ranging from Pluto’s mass (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='002  M⊕) to a hypothetical giant planet’s core (10 M⊕), placed  between 1 and 30 AU where ambient temperatures are too  low to create CSs in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' The grains abundance Zgr  ranges from subsolar (10−3) to supersolar (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  Our results are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' In this plot we  show only the areas of parameter space leading to envelopes  conducive to the creation of CSs and that are self-consistent  to our model assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' This is defined by these condi-  tions:  (i) τvap,sil is less than τmix,d at the base of the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  This simply constrain the envelopes to those where solid  silicates at their base can get vaporized faster than they are  transported back into the upper cooler zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  (ii) The envelope is fully convective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' This ensures that the  newly created CSs can be convectively diffused all the way  back into the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' This condition is inspired by the results of  Ali-Dib & Thompson (2020) who considered a similar setup  with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='3 M⊕ core embedded in the disk, and showed that,  for typical accretion rates, pebble fragmentation and dust  loading increases the opacity and push the convective zone  out till it reaches the Bondi radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Dust particles in these  steady-state envelopes are then diffusively ejected back to  the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Our results rely on this mechanism to transport  the newly created CSs from the hot inner envelope back to  the disk to be incorporated in proto-comets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  (iii) The collisional velocity is higher than 100 cm/s  throughout the envelope, and conditions 11 and 12 are sat-  isfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' This ensures that our dust size prescription is self-  consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='1  Core mass  Figure 1 shows that, while CSs can be created under a va-  riety of parameter ranges, trends do exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' We start with  our nominal model, for  ˙Macc = 10−6M⊕/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' First, there  is a relatively narrow range of masses that extends from  around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='08 M⊕ (40 times Pluto’s mass) to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='5 M⊕, beyond  which the chances of creating CSs drops drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' This  implies that CSs might have formed in the proto-envelopes  of Mars to Earth mass protoplanets that have since disap-  peared via giant collisions or dynamical ejection, or possibly  grown into giant planetary cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' The lower limit on core  masses is mainly due to their envelopes’ relatively cooler  temperatures, increasing τvap,sil considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' On the other  hand, cores with masses higher than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='5 M⊕ have dust par-  ticles large enough in their middle and inner envelopes to  switch from the Rosseland mean opacity regime to the ge-  ometric opacity regime, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 2 (left hand  panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' This decreases the radiative gradient, creating an in-  ner radiative zone that prevents these envelopes from being  fully convective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' It is worth noting that, while our model  considers the smallest of the Hill and Bondi radii to be the  envelope’s outer boundary, all of our acceptable cases that  form CSs are in the Bondi regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' This is expected as the  Hill regime dominates for higher mass cores (5-10 M⊕) that  were excluded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' The Bondi radius RB = 2GM/c2  s is  obtained by equating the local sound speed to the gravi-  tational escape velocity, and thus describes a usually light  but bound envelope where gas particles do not have enough  thermal energy to escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' For higher mass cores, the Bondi  radius is large enough for the Hill stability criteria to become  the more stringent constrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='2  Semimajor axis  A complementary piece of information is the semimajor axis,  where we find that CSs can form almost anywhere in the disk  if the envelope’s grain abundance is high enough as discussed  further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Semimajor axis controls the temperature and  density at the outer boundary, which seems to be important  only in the marginal cases, for example for low core masses  where the envelopes would be too cold if placed further out  in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' The wide range of possible semimajor axis allows  for the possibility of creating CSs in the comets formation  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Classically, Oort cloud comets were thought to form  among the giant planets all the way down to 5 AU, while  Jupiter family comets were thought to form in the scattered  disk (Duncan, Quinn, & Tremaine 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Duncan & Levison  1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Dones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Alternatively, Brasser & Morbidelli  (2013) proposed that both could have formed in the same  region beyond Neptune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='3  Grains abundance  We moreover find that creating CSs necessitate solar to su-  persolar grain abundance in the envelope (Zgr >= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  This result is not consistent with the subsolar grain abun-  dances found in models that incorporate dust growth & set-  tling to the core but omit dust fragmentation with convec-  tive mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Ormel (2014) for example added a simple grain  growth equation to the atmospheric structure equations, and  found that Zgr can be as low as 10−4 in parts of the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  Mordasini (2014) also created an atmospheric model incor-  porating dust settling and coagulation, and found that this  mostly results in subsolar opacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' The main role of Zgr is  MNRAS 000, 1–5 (2023)  4  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Ali-Dib  to increase the opacity and extend the convective zone all the  way to the outer boundary (Bondi or Hill radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Ali-Dib  & Thompson (2020) discussed the gradual buildup of Zgr in  the envelope through accretion and fragmentation, and de-  rived a lower limit on Zgr in order to get a fully convective  envelope:  Zgr > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='12 T 2  d,2  ρd,−11  �tacc,c  Myr  � �  Mc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='3M⊕  �−2/3  (21)  which is generally consistent with our Zgr values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' In order to  get supersolar Zgr, multiple conditions need to be satisfied:  The dust size need to be fragmentation-limited, which  is a pre-requisite for our dust-size prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' This depends  on many factors, including the accretion rate (setting the lu-  minosity and thus convective speeds) and particles’ porosity  and chemical composition (Blum & Wurm 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Okuzumi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Wada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 2008, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  A significant fraction of the dust should not get accreted  by the core, but remain mixed in the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' This is an  open question with many complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' In our cases, sili-  cates are in vapor form at the base of the envelope which  should stop accretion from taking place unless the tempera-  ture is low enough for the inner envelope to reach saturation  pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' This also depends on the nature of convection in  these envelopes, whether it is diffusive as we are assuming,  or whether it is dominated by large scale eddies that can  enhance accretion by the core (Johansen & Nordlund 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='4  Accretion rate  Finally we investigate the effects of using a lower accretion  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Our results for  ˙Macc = 10−7M⊕/yr are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' In this case we find that while the semimajor axis range  remains the same and the lower mass limit does not change  (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='08M⊕), the upper limit decreases by over a factor 2 to  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='6M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' This is expected since, lower accretion rate leads  to lower luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 2 (right hand side),  this decreases the radiative gradient and allows for a radia-  tive zone in the inner envelope even though the dust size in  the 2 cases converge to the same inner value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' In some cases  Zgr can compensate for the lower luminosity and increases  the opacity enough to create fully convective envelopes, ex-  plaining the overall larger Zgr we find for the lower accretion  rate cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  4  SUMMARY & CONCLUSIONS  Crystalline silicates are ubiquitous in comets, but can only  form at very high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Here we investigated the  possibility of transforming amorphous silicates into crys-  talline particles inside the envelopes of protoplanets through  vaporization followed by re-condensation, and then ejecting  them back to the disk through diffusion in the fully con-  vective envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Using a simplified 1D envelope structure  model that incorporates a dust size prescription accounting  for fragmentation and growth, we showed that crystalline  silicates can be created from a diverse set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  Cores need to be between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='08 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='5 M⊕ in mass, as lighter  cores do not allow for temperatures high enough to vaporize  silicates, and the envelopes of more massive cores are often  not fully convective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' We finally found that the location in  the disk (1 to 30 AU) has little influence on the results, ex-  cept in marginal cases, and that a solar to supersolar grain  abundance is needed, but this can be achieved through dust  fragmentation and accumulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Our mechanism is simple  and does not rely on assumptions about the disk, although  it depends on the assumed diffusive nature of 1D convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  Whether this is realistic needs to be investigated further us-  ing 3D hydrodynamic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  We thank the anonymous referee for their constructive com-  ments that greatly improved this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' This work is  supported by Tamkeen under the NYU Abu Dhabi Research  Institute grant CAP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  DATA AVAILABILITY  The data underlying this article (numerical simulations out-  put files) will be shared on reasonable request to the corre-  sponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
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+page_content=', 1994,  Icar, 112, 490  MNRAS 000, 1–5 (2023)  Origins of hot minerals  5  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' The semimajor axis, core mass, and envelope grain abundance for all the cases that satisfy our conditions to form and eject  crystal silicates as enumerated in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Left:  ˙Macc = 10−6M⊕/yr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Right:  ˙Macc = 10−7M⊕/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Note the different color scales for  the two panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  109  1010  1011  r [cm]  10-4  10-3  10-2  10-1  100  101  102  103  104  105  2π ad / λmax  2π ad / λmax,  5M⊕  2π ad / λmax, 1M⊕  Geometric opacity limit  10-1  100  101  102  Temperature gradient  ∇rad, 1M⊕  ∇rad, 5M⊕  ∇ad  109  1010  1011  r [cm]  10-6  10-5  10-4  10-3  10-2  10-1  Dust size [cm]  ad, ˙M=10−6  ad, ˙M=10−7  10-1  100  101  102  Temperature gradient  ∇rad, ˙M=10−6  ∇rad, ˙M=10−7  ∇ad  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Left: solid lines are the opacity efficiency factors Qe (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' 7) for 2 different core masses with all other parameters being equal  (15 AU,  ˙Macc = 10−6M⊕/yr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' These reach the regime switch value of 2 (solid blue line) at different radii, creating a radiative zone in  the inner envelope for the 5 M⊕ case but not for 1 M⊕ due to its smaller dust size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' The dashed lines are the radiative and adiabatic  gradients, indicating the radiative and convective zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Right: solid lines are the dust size ad for cases with 2 different accretion rates  but all other parameters being equal (15 AU, 1 M⊕).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=' Dashed lines are the radiative and adiabatic gradients for the same cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content=', 1999, ApJ, 517,  1034  This paper has been typeset from a TEX/LATEX file prepared by  the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='  MNRAS 000, 1–5 (2023)  Macc = 10-6 Ma /yr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='2  100  Core Mass [M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='0  grai  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='2  6  10-1  Lo  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='6  0  5  10  15  20  25  30  Semimajor axis [AU]Macc = 10-7 Mg / yr  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='16  b-  N  Core Mass [M]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='24  abundance  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='40  og grain  10-1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}
+page_content='72  0  5  10  15  20  25  30  Semimajor axis [AU]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAzT4oBgHgl3EQfgvwn/content/2301.01472v1.pdf'}

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+     
+Compliance Costs of AI Technology 
+Commercialization: A Field Deployment 
+Perspective 
+ 
+Weiyue Wu and Shaoshan Liu 
+ 
+1. Introduction 
+ 
+While Artificial Intelligence (AI) technologies are progressing fast, compliance costs have 
+become a huge financial burden for AI startups, which are already constrained on research & 
+development budgets. This situation creates a compliance trap, as many AI startups are not 
+financially prepared to cope with a broad spectrum of regulatory requirements. Particularly, 
+the complex and varying regulatory processes across the globe subtly give advantages to well-
+established and resourceful technology firms over resource-constrained AI startups [1]. The 
+continuation of this trend may phase out the majority of AI startups and lead to giant 
+technology firms' monopolies of AI technologies. To demonstrate the reality of the 
+compliance trap, from a field deployment perspective, we delve into the details of compliance 
+costs of AI commercial operations. 
+2. Financial Vulnerability: Tech Giants vs. AI Startups 
+ 
+Compared to established tech giants, AI startups are much more financially vulnerable. 
+Based on the OECD Regulatory Compliance Cost Assessment Guidance [2], we quantitatively 
+compare the financial vulnerability of tech giants versus AI startups. Conducting a financial 
+statement simulation provides a glimpse at the impact of changes in compliance costs. The 
+simulation in Table 1 assumes that gross profit and expense are a fixed percentage of revenue. 
+The compliance cost is split into a fixed cost regardless of revenue and a variable cost that 
+takes 5% of revenue. When the fixed compliance cost increases by 200%, the operating 
+margin of the startup changes from 13% to -7%, turning the company from a profitable 
+business into a money-losing one. By contrast, such a change only brings a slight drop in 
+operating margin in tech giants.  
+ 
+Table 1: Sensitivity Analysis: Impact of Change in Compliance Cost on Operating Income 
+
+Start-ups
+Tech Giants
+200% increase in
+200% increase in
+Base case
+fixed compliance cost
+Base case
+fixed compliance cost
+Revenue
+$2,000,000
+$2,000,000
+$20,000,000
+$20,000,000
+Gross Margin
+40%
+40%
+40%
+40%
+Gross Profit
+800,000
+800,000
+8,000,000
+8,000,000
+Total Compliance Cost
+300,000
+700,000
+1,200,000
+1,600,000
+Fixed compliance cost
+200,000
+600,000
+200,000
+600,000
+Variable compliance cost
+100,000
+100,000
+1,000,000
+1,000,000
+Expenses
+240,000
+240,000
+2,400,000
+2,400,000
+Operating Income
+260,000
+(140,000)
+4,400,000
+4,000,000
+Operating Margin
+13%
+-7%
+22%
+20% 
+The actual situation of AI startups is more complex than this estimation, as it is nearly 
+impossible for external analysts to estimate the compliance costs. According to Accounting 
+Standards, companies are not required to disclose compliance cost explicitly in financial 
+reports when the cost is not material [3]. Thus, compliance costs may become hidden by 
+nature and classified into other categories, such as Research and Development expenses and 
+other administrative expenses. Lacking first-hand information, analysts on the macro-
+economic level tend to underestimate the costs of AI regulations. For instance, in an impact 
+assessment report of the Europe AI Act, the estimated annual compliance cost of one AI 
+product that averagely costs EUR 170,000 to develop is EUR 29,277, we believe this study has 
+underestimated the actual costs of AI compliance [4].   
+ 
+3. The Compliance Trap 
+ 
+AI is a highly regulated industry, but unfortunately, there is no standardized AI regulation 
+framework, and hence compliance costs often become a financial trap for AI startups [1]. 
+Most AI entrepreneurs may not even be aware of the existence of compliance costs, let alone 
+the severe impact compliance costs may have on the company's overall financial health. As 
+illustrated in Figure 1, we summarize the following challenges.   
+ 
+ 
+Figure 1: the compliance trap for AI startups 
+ 
+First, unlike R&D budgeting, due to varying AI regulatory frameworks across the globe or 
+even across multiple regions within a country, there is no standard method to budget for AI 
+compliance costs. Even estimating the range of AI compliance costs is infeasible.  
+ 
+Second, even with an AI compliance budget, the actual costs may significantly deviate from 
+the budget. AI startups often encounter new compliance issues as they progress through 
+commercialization. In addition, opportunity costs arise as regulators inspect AI products on 
+safety and privacy issues, causing delays in commercial deployments. 
+ 
+Third, varying AI regulations often introduce indirect costs. For instance, a strict compliance 
+environment demands engineers deal with regulatory issues such as responding to various 
+compliance technical inquiries instead of spending time developing products. Such a shift of 
+focus does not reflect in financial reports, as engineers' costs are categorized as R&D costs.  
+ 
+
+No Standard Method to Budget
+Deviation from Compliance Budget
+Indirect Costs
+Al start-upsfacevarying Al regulatory
+Al start-ups encounter new compliance issues and
+Al start-ups might work around compliance
+frameworks across theglobe
+delay of commercial deployments
+by other activities such as R&D and marketing
+Compliancebudgetplanningoftechcompanies
+Rough budget-overrun disaggregation,%
+Classification ofCompliance Costs
+- Standard waterfall method
+Pro-forma
+Certification cost
+Situation
+Revise
+Finalize
+budget
+budget
+budget
+Regulatory complexity
+orecast
+Actual
+CompliancebudgetplanningofAl companies
+Shifting requirements
+cost
+Marketing cost
+Total Cost
+R&Dcost
+- On an Ad-Hoc basis
+of
+50
+Lack of standards
+Compliance
+Logistics cost
+Training cost
+Testing and
+Qualityassurance
+100
+Insurancecost
+Budget
+30
+validation cost
+cost
+Documentation
+Marketing cost
+cost
+15
+Labor cost
+Direct cost
+Indirect cost4. A Field Deployment Perspective 
+ 
+In this section, with more than six years of first-hand experience in deploying commercial 
+autonomous driving services, we delve into the details of compliance costs from a field 
+deployment perspective, in the hope that the insights we provide can raise awareness of the 
+adverse impact of the lack of standardized AI regulations.  
+ 
+4.1. Background 
+ 
+    PerceptIn is an autonomous driving startup founded in 2016. It offers autonomous micro-
+mobility solutions to customers from the United States, European Union, and Asia. The 
+company only budgeted for ordinary compliance expenses, such as the direct labor cost of a 
+safety driver on board and the equipment cost of a waterproof surveillance camera. While 
+facing a broad spectrum of regulatory obstacles across different countries, PerceptIn had 
+fallen into the compliance trap. Many financial and human resources have been spent out of 
+budget to comply with various regulatory frameworks in different regions.  
+ 
+4.2. Scenario 1 – No Standard Method to Budget 
+     
+The AI regulation framework in China was blurry, and when the company first launched the 
+autonomous micro-mobility project in China, it was impossible to budget for compliance costs. 
+For instance, since relevant regulations were absent back then, the company needed to 
+develop its own testing plan to obtain deployment approval. Without detailed testing 
+standards, the company had to spend $25,000 per month to simulate real-world scenarios at 
+the initial stage for a testing site for testing and demonstration purposes. The testing process 
+is to obtain detailed validation results, whereas the demonstration process is to convince the 
+regulatory body regarding the safety and reliability of the service. The $300,000 annual cost 
+was not included in the company’s original budget and imposed a heavy burden on the 
+company's financial health. 
+ 
+4.3. Scenario 2 – Deviation from Compliance Budget 
+ 
+The company was invited to launch an autonomous driving pilot program in a European 
+city. Before rolling out the project, the company was asked to prepare a risk mitigation plan 
+for 40 different scenarios. To cope with the regulatory process, the R&D team shifted its focus 
+to responding to scenarios-based functional specifications and supplemented the mitigation 
+plan with real-time data. During the project budgeting phase, the company had prepared 20 
+man-days to cope with the AI regulatory process. Nevertheless, the process turned out to 
+consume 400 man-days to complete the process. While the original budget was $10,000, in 
+the end, the process consumed $200,000. Such a severe mismatch was caused by the lack of 
+a standardized regulatory process, as any response from our technical team would bring on 
+the next round of regulatory questions.  
+ 
+4.4. Scenario 3 – Indirect Costs  
+     
+
+Japan is famous for its rigid structure in organizations. Thus without an established 
+compliance process in place, gaining the confidence of the Japanese government is essential 
+to commercial deployment. To gain the confidence of the Japanese government, the company 
+first debuted a marketing campaign to promote safe autonomous micro-mobility services in 
+a smart city project [5]. With a successful local case and globally established brand, the 
+company then discussed operation permits with the Ministry of Land, Infrastructure, 
+Transport, and Tourism (MLIT) [6]. The preparation and initiation of the project took over 24 
+months, costing $500,000 in promotion, material preparation, and marketing campaigns. 
+Traditionally marketing activities were not meant to cope with compliance requirements. 
+However, in this case, marketing was a tool to convince the regulatory body to further 
+autonomous driving operation permits.  
+ 
+In the case of PerceptIn, the compliance cost of one deployment project is $ 344,000 on 
+average, whereas the average R&D cost is around $150,000, making the compliance costs 2.3 
+times the amount of R&D costs, far exceeding the 17.6% estimation of the Europe AI Act. 
+ 
+5. The Silver Linings 
+ 
+The root cause of the compliance trap is the lack of a standardized AI regulatory framework. 
+An ultimate solution to this problem lies in creating a global golden standard for AI regulation. 
+A study of the Food and Drug Administration's (FDA's) history sheds light on properly 
+regulating a new field. First, pharmaceutical products a century ago and AI today are both 
+viewed as black boxes. Even the most sophisticated scientists could not predict their 
+development trajectory, let alone legal experts. Second, pharmaceutical and AI technologies 
+can potentially cause severe public risks if they are not adequately regulated. Third, both 
+industries have enormous potential for improving people's well-being. Like the FDA, a 
+consumer protection agency shall be established to ensure that AI is developed for people's 
+well-being. Such an agency should develop the expertise and capability to scientifically judge 
+whether an AI product is ethical and legal. Such an agency should provide guidance to various 
+governments within the U.S. and worldwide on AI regulations.  
+However, before a consensus can be reached regarding the golden standard, a new 
+business model, Compliance-as-a-Service (CaaS), can specialize in dealing with varying AI 
+regulatory frameworks and thus amortize compliance costs across different AI startups. In 
+addition, CaaS reduces the friction between regulatory bodies and AI startups by providing an 
+interface to compile legal terms into technical and operational plans. With the new business 
+model, AI entrepreneurs can adequately budget for compliance when evaluating the potential 
+of an innovative idea. 
+ 
+6. Summary 
+ 
+AI is a promising industry mainly filled with startups exploring the applications of AI 
+technologies in different aspects of our daily lives.  Compared to well-established tech giants, 
+AI startups are financially vulnerable. Unfortunately, the lack of standardized AI regulatory 
+frameworks creates a compliance trap that may destroy an AI startup financially, which could 
+lead to a more profound impact of creating a competitive advantage for tech giants over AI 
+
+startups. We have examined the details of compliance costs from a field deployment 
+perspective to demonstrate the reality of the compliance trap. Ideally, if a global golden 
+standard on AI regulation could be developed, then AI startups could accurately budget for 
+compliance costs. However, before a consensus can be reached regarding the golden 
+standard, we believe that a new business model, compliance as a service, can specialize in 
+dealing with varying AI regulatory frameworks and thus amortize compliance costs across 
+different AI startups.  
+ 
+References: 
+1. Wu, W. and Liu, S., 2021. Dilemma of the Artificial Intelligence Regulatory 
+Landscape. Communications of the ACM, 2023. 
+2. OECD. Publishing, 2014. OECD Regulatory Compliance Cost Assessment Guidance. OECD Publishing. 
+3. PricewaterhouseCoppers, 2022. Illustrative IFRS consolidated financial statements. 
+4. Renda, A., Arroyo, J., Fanni, R., Laurer, M., Sipiczki, A., Yeung, T., Maridis, G., Fernandes, M., Endrodi, 
+G. and Milio, S., 2021. Study to support an impact assessment of regulatory requirements for artificial 
+intelligence in Europe. European Commission: Brussels, Belgium. 
+5. Fukuoka City conducts demonstration test of compact self-driving car by US company PerceptIn, Inc.. 
+Nikkei, accessed 2023-01-05, https://www.nikkei.com/article/DGXLRSP518592_W9A900C1000000/ 
+6. List of Proposal Sectors and Private Companies, etc. . In Seeds proposal for realization of smart island, 
+Japanese Ministry of Land, Infrastructure, Transport and Tourism, accessed 2023-01-05, 
+https://www.mlit.go.jp/kokudoseisaku/chirit/kokudoseisaku_chirit_tk_000309.html 
+ 
+ 
+Biography: 
+ 
+Weiyue Wu is Chief Operating Officer of PerceptIn, an autonomous driving startup founded 
+in 2016. At PerceptIn, she has been in charge of commercial autonomous driving service 
+deployments in the US, Europe, Japan, and China. Before PerceptIn, she served as Investment 
+Director of Oxford Seed Fund and Investment Advisor of ARM Accelerator. She began her 
+career as a Multi-National Corporation Compliance Auditor at KPMG and a Senior Automobile 
+Consultant at Deloitte. She received her MBA from the University of Oxford. She is a founding 
+member of IEEE Special Technical Community on Autonomous Driving Technologies, a 
+Certified Public Accountant and a practicing lawyer in China. 
+ 
+Dr. Shaoshan Liu’s background is a unique combination of technology, entrepreneurship, and 
+public policy, which enables him to take on great global challenges.  On technology, Dr. Liu 
+has published 4 textbooks, more than 100 research papers, and holds more than 150 patents 
+in autonomous systems. On entrepreneurship, Dr. Shaoshan Liu is CEO of PerceptIn and has 
+commercially deployed autonomous micro-mobility services in the U.S., Europe, Japan, and 
+China etc. He is the Asia Chair of IEEE Entrepreneurship. On public policy, Dr. Liu has served 
+on the World Economic Forum’s panel on Industry Response to Government Procurement 
+Policy, is leading the Autonomous Machine Computing roadmap under IEEE International 
+Roadmap of Devices and Systems (IRDS) and is a member of the ACM U.S. Technology Policy 
+Committee. Dr. Liu’s educational background includes a M.S. in Biomedical Engineering, a 
+Ph.D. in Computer Engineering from the U.C. Irvine, and a Master of Public Administration 
+(MPA) from Harvard University. He is an IEEE Senior Member, an IEEE Computer Society 
+Distinguished Speaker, an ACM Distinguished Speaker, an Advisory Council member of 
+
+Harvard Business Review, a member of MIT Technology Review’s Global Insights Panel, and a 
+member of the Forbes Technology Council.  
+

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+page_content='Compliance Costs of AI Technology Commercialization: A Field Deployment Perspective  Weiyue Wu and Shaoshan Liu  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Introduction  While Artificial Intelligence (AI) technologies are progressing fast, compliance costs have become a huge financial burden for AI startups, which are already constrained on research & development budgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' This situation creates a compliance trap, as many AI startups are not financially prepared to cope with a broad spectrum of regulatory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Particularly, the complex and varying regulatory processes across the globe subtly give advantages to well- established and resourceful technology firms over resource-constrained AI startups [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=" The continuation of this trend may phase out the majority of AI startups and lead to giant technology firms' monopolies of AI technologies." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' To demonstrate the reality of the compliance trap, from a field deployment perspective, we delve into the details of compliance costs of AI commercial operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Financial Vulnerability: Tech Giants vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' AI Startups  Compared to established tech giants, AI startups are much more financially vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Based on the OECD Regulatory Compliance Cost Assessment Guidance [2], we quantitatively compare the financial vulnerability of tech giants versus AI startups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Conducting a financial statement simulation provides a glimpse at the impact of changes in compliance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' The simulation in Table 1 assumes that gross profit and expense are a fixed percentage of revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' The compliance cost is split into a fixed cost regardless of revenue and a variable cost that takes 5% of revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' When the fixed compliance cost increases by 200%, the operating margin of the startup changes from 13% to -7%, turning the company from a profitable business into a money-losing one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' By contrast, such a change only brings a slight drop in operating margin in tech giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  Table 1: Sensitivity Analysis: Impact of Change in Compliance Cost on Operating Income  Start-ups Tech Giants 200% increase in 200% increase in Base case fixed compliance cost Base case fixed compliance cost Revenue $2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 $2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 $20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 $20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 Gross Margin 40% 40% 40% 40% Gross Profit 800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 800,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 Total Compliance Cost 300,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 700,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='600,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 Fixed compliance cost 200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 600,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 200,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 600,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 Variable compliance cost 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 Expenses 240,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 240,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='400,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='400,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 Operating Income 260,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 (140,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000) 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='400,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='000 Operating Margin 13% -7% 22% 20% The actual situation of AI startups is more complex than this estimation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' as it is nearly impossible for external analysts to estimate the compliance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' According to Accounting Standards, companies are not required to disclose compliance cost explicitly in financial reports when the cost is not material [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Thus, compliance costs may become hidden by nature and classified into other categories, such as Research and Development expenses and other administrative expenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Lacking first-hand information, analysts on the macro- economic level tend to underestimate the costs of AI regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' For instance, in an impact assessment report of the Europe AI Act, the estimated annual compliance cost of one AI product that averagely costs EUR 170,000 to develop is EUR 29,277, we believe this study has underestimated the actual costs of AI compliance [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' The Compliance Trap  AI is a highly regulated industry, but unfortunately, there is no standardized AI regulation framework, and hence compliance costs often become a financial trap for AI startups [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=" Most AI entrepreneurs may not even be aware of the existence of compliance costs, let alone the severe impact compliance costs may have on the company's overall financial health." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' As illustrated in Figure 1, we summarize the following challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  Figure 1: the compliance trap for AI startups  First, unlike R&D budgeting, due to varying AI regulatory frameworks across the globe or even across multiple regions within a country, there is no standard method to budget for AI compliance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Even estimating the range of AI compliance costs is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  Second, even with an AI compliance budget, the actual costs may significantly deviate from the budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' AI startups often encounter new compliance issues as they progress through commercialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' In addition, opportunity costs arise as regulators inspect AI products on safety and privacy issues, causing delays in commercial deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  Third, varying AI regulations often introduce indirect costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' For instance, a strict compliance environment demands engineers deal with regulatory issues such as responding to various compliance technical inquiries instead of spending time developing products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=" Such a shift of focus does not reflect in financial reports, as engineers' costs are categorized as R&D costs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  No Standard Method to Budget Deviation from Compliance Budget Indirect Costs Al start-upsfacevarying Al regulatory Al start-ups encounter new compliance issues and Al start-ups might work around compliance frameworks across theglobe delay of commercial deployments by other activities such as R&D and marketing Compliancebudgetplanningoftechcompanies Rough budget-overrun disaggregation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='Classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='ofCompliance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
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+page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='Lack ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='standards ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='Compliance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='Logistics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='Training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='Testing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='Qualityassurance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='Insurancecost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='Budget ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='validation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='Documentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='Marketing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='Labor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='Direct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='Indirect ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='cost4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' A Field Deployment Perspective  In this section, with more than six years of first-hand experience in deploying commercial autonomous driving services, we delve into the details of compliance costs from a field deployment perspective, in the hope that the insights we provide can raise awareness of the adverse impact of the lack of standardized AI regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Background  PerceptIn is an autonomous driving startup founded in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' It offers autonomous micro- mobility solutions to customers from the United States, European Union, and Asia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' The company only budgeted for ordinary compliance expenses, such as the direct labor cost of a safety driver on board and the equipment cost of a waterproof surveillance camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' While facing a broad spectrum of regulatory obstacles across different countries, PerceptIn had fallen into the compliance trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Many financial and human resources have been spent out of budget to comply with various regulatory frameworks in different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Scenario 1 – No Standard Method to Budget  The AI regulation framework in China was blurry, and when the company first launched the autonomous micro-mobility project in China, it was impossible to budget for compliance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' For instance, since relevant regulations were absent back then, the company needed to develop its own testing plan to obtain deployment approval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Without detailed testing standards, the company had to spend $25,000 per month to simulate real-world scenarios at the initial stage for a testing site for testing and demonstration purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' The testing process is to obtain detailed validation results, whereas the demonstration process is to convince the regulatory body regarding the safety and reliability of the service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=" The $300,000 annual cost was not included in the company’s original budget and imposed a heavy burden on the company's financial health." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Scenario 2 – Deviation from Compliance Budget  The company was invited to launch an autonomous driving pilot program in a European city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Before rolling out the project, the company was asked to prepare a risk mitigation plan for 40 different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' To cope with the regulatory process, the R&D team shifted its focus to responding to scenarios-based functional specifications and supplemented the mitigation plan with real-time data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' During the project budgeting phase, the company had prepared 20 man-days to cope with the AI regulatory process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Nevertheless, the process turned out to consume 400 man-days to complete the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' While the original budget was $10,000, in the end, the process consumed $200,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Such a severe mismatch was caused by the lack of a standardized regulatory process, as any response from our technical team would bring on the next round of regulatory questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Scenario 3 – Indirect Costs  Japan is famous for its rigid structure in organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Thus without an established compliance process in place, gaining the confidence of the Japanese government is essential to commercial deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' To gain the confidence of the Japanese government, the company first debuted a marketing campaign to promote safe autonomous micro-mobility services in a smart city project [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' With a successful local case and globally established brand, the company then discussed operation permits with the Ministry of Land, Infrastructure, Transport, and Tourism (MLIT) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' The preparation and initiation of the project took over 24 months, costing $500,000 in promotion, material preparation, and marketing campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Traditionally marketing activities were not meant to cope with compliance requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' However, in this case, marketing was a tool to convince the regulatory body to further autonomous driving operation permits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  In the case of PerceptIn, the compliance cost of one deployment project is $ 344,000 on average, whereas the average R&D cost is around $150,000, making the compliance costs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='3 times the amount of R&D costs, far exceeding the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='6% estimation of the Europe AI Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' The Silver Linings  The root cause of the compliance trap is the lack of a standardized AI regulatory framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' An ultimate solution to this problem lies in creating a global golden standard for AI regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=" A study of the Food and Drug Administration's (FDA's) history sheds light on properly regulating a new field." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' First, pharmaceutical products a century ago and AI today are both viewed as black boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Even the most sophisticated scientists could not predict their development trajectory, let alone legal experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Second, pharmaceutical and AI technologies can potentially cause severe public risks if they are not adequately regulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=" Third, both industries have enormous potential for improving people's well-being." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=" Like the FDA, a consumer protection agency shall be established to ensure that AI is developed for people's well-being." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Such an agency should develop the expertise and capability to scientifically judge whether an AI product is ethical and legal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Such an agency should provide guidance to various governments within the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' and worldwide on AI regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' However, before a consensus can be reached regarding the golden standard, a new business model, Compliance-as-a-Service (CaaS), can specialize in dealing with varying AI regulatory frameworks and thus amortize compliance costs across different AI startups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' In addition, CaaS reduces the friction between regulatory bodies and AI startups by providing an interface to compile legal terms into technical and operational plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  With the new business model, AI entrepreneurs can adequately budget for compliance when evaluating the potential of an innovative idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Summary  AI is a promising industry mainly filled with startups exploring the applications of AI technologies in different aspects of our daily lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Compared to well-established tech giants, AI startups are financially vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Unfortunately, the lack of standardized AI regulatory frameworks creates a compliance trap that may destroy an AI startup financially, which could lead to a more profound impact of creating a competitive advantage for tech giants over AI  startups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' We have examined the details of compliance costs from a field deployment perspective to demonstrate the reality of the compliance trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Ideally, if a global golden standard on AI regulation could be developed, then AI startups could accurately budget for compliance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' However, before a consensus can be reached regarding the golden standard, we believe that a new business model, compliance as a service, can specialize in dealing with varying AI regulatory frameworks and thus amortize compliance costs across different AI startups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  References: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Wu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' and Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Dilemma of the Artificial Intelligence Regulatory Landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Communications of the ACM, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' OECD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Publishing, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' OECD Regulatory Compliance Cost Assessment Guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' OECD Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' PricewaterhouseCoppers, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Illustrative IFRS consolidated financial statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Renda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=', Arroyo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=', Fanni, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=', Laurer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=', Sipiczki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=', Yeung, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=', Maridis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=', Fernandes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=', Endrodi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' and Milio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Study to support an impact assessment of regulatory requirements for artificial intelligence in Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' European Commission: Brussels, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Fukuoka City conducts demonstration test of compact self-driving car by US company PerceptIn, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='. Nikkei, accessed 2023-01-05, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='nikkei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='com/article/DGXLRSP518592_W9A900C1000000/ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' List of Proposal Sectors and Private Companies, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' In Seeds proposal for realization of smart island, Japanese Ministry of Land, Infrastructure, Transport and Tourism, accessed 2023-01-05, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='mlit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='jp/kokudoseisaku/chirit/kokudoseisaku_chirit_tk_000309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='html  Biography:  Weiyue Wu is Chief Operating Officer of PerceptIn, an autonomous driving startup founded in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' At PerceptIn, she has been in charge of commercial autonomous driving service deployments in the US, Europe, Japan, and China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Before PerceptIn, she served as Investment Director of Oxford Seed Fund and Investment Advisor of ARM Accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' She began her career as a Multi-National Corporation Compliance Auditor at KPMG and a Senior Automobile Consultant at Deloitte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' She received her MBA from the University of Oxford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' She is a founding member of IEEE Special Technical Community on Autonomous Driving Technologies, a Certified Public Accountant and a practicing lawyer in China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='  Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Shaoshan Liu’s background is a unique combination of technology, entrepreneurship, and public policy, which enables him to take on great global challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' On technology, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Liu has published 4 textbooks, more than 100 research papers, and holds more than 150 patents in autonomous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' On entrepreneurship, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Shaoshan Liu is CEO of PerceptIn and has commercially deployed autonomous micro-mobility services in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=', Europe, Japan, and China etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' He is the Asia Chair of IEEE Entrepreneurship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' On public policy, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Liu has served on the World Economic Forum’s panel on Industry Response to Government Procurement Policy, is leading the Autonomous Machine Computing roadmap under IEEE International Roadmap of Devices and Systems (IRDS) and is a member of the ACM U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Technology Policy Committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Liu’s educational background includes a M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' in Biomedical Engineering, a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' in Computer Engineering from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' Irvine, and a Master of Public Administration (MPA) from Harvard University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}
+page_content=' He is an IEEE Senior Member, an IEEE Computer Society Distinguished Speaker, an ACM Distinguished Speaker, an Advisory Council member of  Harvard Business Review, a member of MIT Technology Review’s Global Insights Panel, and a member of the Forbes Technology Council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9FQT4oBgHgl3EQf-TfR/content/2301.13454v1.pdf'}

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+arXiv:2301.03105v1  [math.GT]  8 Jan 2023
+FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT
+BUNDLES
+NIMA ANVARI AND IAN HAMBLETON
+Abstract. Given a 4-manifold with a homologically trivial and locally-linear cyclic
+group action, we obtain necessary and sufficient conditions for the existence of equi-
+variant bundles.
+The conditions are derived from the twisted signature formula and
+are in the form of congruence relations between the fixed point data and the isotropy
+representations.
+1. Introduction
+Finite group actions on 4-manifolds can be studied in various settings. We are mainly
+interested in comparing smooth actions with those which are topological and locally linear,
+but important examples arise for symplectic 4-manifolds and complex surfaces. Here is a
+sampling of survey articles and recent work on aspects of this general theme: [1, 2, 3, 5,
+6, 7, 10, 15, 22, 20, 21, 23, 26, 31, 36]. We will focus on the existence and classification
+of equivariant bundles, and their applications in Yang-Mills gauge theory to the study of
+finite group actions.
+We begin by recalling some standard definitions. Let (X, π) denote a simply-connected,
+closed 4-manifold X together with a locally linear and homologically trivial action of a
+cyclic group π = Z/p of prime order. The fixed point set Xπ has Euler characteristic
+χ(Xπ) = b2(X) + 2, by the Lefschetz fixed point formula. In general Xπ will consist of
+isolated fixed points and a disjoint union of fixed 2-spheres (see [9, §2]).
+Definition 1.1. At each isolated fixed point x ∈ Xπ, the tangent space admits an equi-
+variant decomposition (TxiX, π) = C(ai) ⊕ C(bi) of complex representation spaces. Let
+t · (z1, z2) = (ζaiz1, ζbiz2)
+denote the action for t a fixed generator in the cyclic group π, ζ = e2πi/p, and with integers
+(ai, bi), both non-zero modulo p.
+(i) The integers (ai, bi) are the local tangential rotation data, and are well-defined up
+to order and simultaneous change in sign.
+(ii) Similarly, for each point x on a π-fixed 2-sphere Fj, there is a representation
+C ⊕ C(cj) corresponding to the equivariant splitting
+TX | Fj = TFj ⊕ N(Fj)
+where N(Fj) is the normal bundle with rotation ζcj, and cj ̸≡ 0 (mod p).
+Date: January 4, 2023.
+This research was partially supported by NSERC Discovery Grant A4000.
+1
+
+2
+NIMA ANVARI AND IAN HAMBLETON
+(iii) The total fixed point rotation data is the collection
+F = {(ai, bi), (cj, αj) | i ∈ I, j ∈ J}.
+where I and J index the isolated fixed points and 2-spheres respectively and
+αj = [Fj]·[Fj] is the self-intersection number of the fixed spheres [Fj] ∈ H2(X; Z).
+(iv) By an equivariant line bundle (L, π) → (X, π) we mean a principal U(1)-bundle
+L over X together with a lift of the π-action to the total space. Given such a
+lift, there exists a set of isotropy representations of the π-action on each fiber
+over the fixed point set which we denote by L | xi = tλi over isolated fixed points
+and L | Fj = tλj over a fixed 2-sphere. Denote the collection of these isotropy
+representations by I = {λi, λj | i ∈ I, j ∈ J}.
+With this notation we have our main result.
+Theorem A. Let (X, π) denote a simply-connected, closed 4-manifold with a locally linear,
+homologically trivial action of a cyclic group π = Z/p of odd prime order p, with fixed-point
+rotation data F = {(ai, bi), (cj, αj) | i ∈ I, j ∈ J}.
+A collection I of integers {λi, λj | i ∈ I, j ∈ J} can be realized (modulo p) as the isotropy
+representations of an equivariant line bundle (L, π) → (X, π) if and only if there is a
+collection of integers {mj | j ∈ J} such that
+(1.1)
+�
+i∈I
+λi
+aibi
++
+�
+j∈J
+cjmj − λjαj
+c2
+j
+≡ 0
+(mod p).
+When a solution exists, the integers mj = c1(i∗L)[Fj] satisfy equation (1.1).
+Remark 1.2. A special case of this result can be found in [22, Proposition 1.4], for a
+cyclic group π of odd order acting locally linearly and semi-freely on the complex projective
+plane CP 2 with has three isolated fixed points. The necessary condition (1.1) in Theorem
+A is established by extending the methods of [22, §2] to more general actions.
+Remark 1.3. The definitions above generalize directly to actions (X, π) where π is any
+finite group, and the structural group G of the principal bundle is any compact Lie group.
+For applications in gauge theory, G = SU(2) is an important example. The details will
+be left to the reader (see also [18, 19]).
+2. Some motivating questions
+Here are some questions related to the general theme (all actions will be assumed to
+preserve orientation). More information about some of these directions can be found in
+the references.
+1. Does there exist a smooth Z/p-action on a (homotopy) K3 surface, which induces the
+identity on integral homology ?
+This is a well-known question of Allan Edmonds. Note that results of Edmonds and
+Ewing [12] imply that topological locally linear examples exist for odd primes.
+2. Does there exist a smooth Z/p-action on a homotopy K3 surface, which contains an
+invariant embedded Brieskorn homology 3-sphere Σ(2, 3, 7) ?
+
+FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES
+3
+Fintushel and Stern [14] showed that many homotopy K3 surfaces admit embeddings
+of Σ(2, 3, 7). The question concerns the possible existence of an equivariant splitting of
+the K3 surface along the Brieskorn sphere.
+3. A Brieskorn homology 3-sphere Σ(a, b, c) admits a free Z/p-action if p ∤ abc. Does there
+exist a smooth, homologically trivial extension of this action with isolated fixed points to
+any smooth simply connected negative definite 4-manifold X with boundary Σ(a, b, c) ?
+Anvari [1] proved that the free Z/7 on Σ(2, 3, 5) does not extend in this way over the min-
+imal negative definite 4-manifold obtained by resolving the link singularity. The method of
+proof involves studying equivariant Yang-Mills gauge theory on the non-compact manifold
+with cylindrical end obtained from X by attaching the end Σ(2, 3, 5)×[0, ∞). Information
+about the Floer homology of Brieskorn spheres is an essential ingredient in tackling this
+problem (see Saveliev [32, 33, 34]).
+4. What sets of rotation numbers can be realized by a smooth, pseudo-free Z/n-action
+on X = S2 × S2 ?
+A pseudo-free action is one with isolated singular points. If the action is semi-free,
+there are “standard models” with rotation data {(a, b), (c, d), (a, −b), (c, −d)} at the four
+fixed points. This question for X = CP 2 was answered in [11, 20], where it turned out
+that the rotation data was the same for locally linear actions.
+5. Let (X, π) denote a smooth action of a finite group π on a closed, simply connected
+smooth 4-manifold. Under what conditins does there exist a π-equivariant principal G-
+bundle over X with prescribed Chern classes, for G = U(1) or G = SU(2) ?
+Some information about this question was provided in [22, 20] for X = CP 2 and π
+finite cyclic, or more generally for X negative definite (see also [19] for the connection
+between Chern classes and the isotropy representations).
+For X = S4, the existence
+and classification of such bundles was applied by Austin [4] and Furuta [15, 17, 16] to
+study group actions via instanton gauge theory. The compactification of an equivariant
+version of Donaldson’s Yang-Mills moduli space [8], [20] involves “bubbling” convergence
+to equivariant instantons over the 4-sphere. Further applications of equivariant bundles
+arise in studying the equivariant compactification of moduli spaces over cylindrical end
+4-manifolds.
+3. The G-Signature Formula
+In this section we review the terms of the G-signature formula that we will need in
+deriving congruence relations.
+The G-signature of a closed 4-manifold X with an orientation preserving action of a
+finite group G acting as isometries on X is defined as the virtual representation
+(3.1)
+Sign(X, G) = [H2
++(X; C)] − [H2
+−(X; C)]
+where H2
+±(X; C) are the maximal positive/negative definite G-invariant subspaces of
+H2(X; C). Taking characters gives the g-signatures
+(3.2)
+Sign(g, X) = trg H2
++(X) − trg H2
+−(X)
+
+4
+NIMA ANVARI AND IAN HAMBLETON
+which by the G-signature formula can be computed from the fixed point set Xg.
+Let D : C∞(Λ+) → C∞(Λ−) denote the signature operator. The Lefschetz numbers
+are computed as follows.
+(3.3)
+L(g, D) = (−1)n(n+1)/2 chg(Λ+ − Λ−)(TX | Xg ⊗ C)Td(TXg ⊗ C)
+e(TXg) chg(Λ−1Ng ⊗ C)
+[Xg]
+Where n = dim Xg and note that TX | Xg = TXg ⊕ Ng and
+(3.4)
+chg(Λ+ − Λ−)(TX | Xg ⊗ C) = chg(Λ+ − Λ−)(TXg ⊗ C) chg(Λ+ − Λ−)(Ng ⊗ C).
+Let F denote a fixed surface, then the contribution to the g-signature is given by
+L(g, D) | F = (−1)(e−x − ex)(e−y−iθ − ey+iθ)
+x(1 − ey+iθ)(1 − e−y−iθ)
+x(−x)
+(1 − e−x)(1 − ex)[F]
+= coth
+�y + iθ
+2
+�
+x cot
+�x
+2
+�
+[F]
+where the following trigonometric identity is used
+(3.5)
+coth
+�x
+2
+�
+=
+e−x − ex
+(1 − e−x)(1 − ex).
+To evaluate on [F], we use the Taylor expansions
+x coth (x/2) = 2 + 1/6x2 + · · ·
+coth
+�y + iθ
+2
+�
+= coth (iθ/2) − 1
+2 csch2
+�iθ
+2
+�
+y
+Thus the contribution to the Lefschetz number is given by
+L(g, D) | F = {2 coth(iθ/2) − csch2(iθ/2)y}[F]
+= − csch2(iθ/2)[F]2 = csc2(θ/2)[F]2
+=
+−4tcF
+(tcF − 1)2[F]2
+where θ = 2πcF
+p
+and cF is the rotation number on the normal fiber of F and t = e2πi/p is
+a primitive pth root of unity. Similarly we can compute the contribution from isolated
+fixed points:
+L(g, D) | pt =
+(e−iθ1 − eiθ1)(e−iθ2 − eiθ2)
+(1 − eiθ1)(1 − e−iθ1)(1 − eiθ2)(1 − e−iθ2)
+= coth(iθ1/2) coth(iθ2/2)
+= − cot(θ1/2) cot(θ2/2)
+= (ta + 1)(tb + 1)
+(ta − 1)(tb − 1)
+
+FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES
+5
+where θ1 = 2πa
+p
+and θ2 = 2πb
+p
+are the rotation numbers at the fixed point. By summing
+over the fixed point set the G-signature formula is given by
+(3.6)
+Sign(X) =
+�
+i
+(tai + 1)
+(tai − 1)
+(tbi + 1)
+(tbi − 1) +
+�
+j
+−4αjtcj
+(tcj − 1)2
+where αj denotes the self-intersection [Fj] · [Fj] of the fixed 2-spheres {Fj}. This formula
+can be viewed as an equation in the cyclotomic field Q[ζ] = Q[t]/Φp(t) where Φp(t) is the
+cyclotomic polynomial 1 + t + t2 + · · · + tp−1.
+4. Congruence Relations for the G-Signature Formula
+In this section we derive congruence relations satisfied by the rotation data F using the
+G-signature formula. We first note that since (ta−1)/(t−1) is a unit in ring of cyclotomic
+integers Z[ζ] when (a, p) = 1, multiplying both sides of the G-signature formula by (t−1)2
+induces an equation in the ring R = Z[ζ]. The I-adic expansion of the resulting right-
+hand side leads to congruence relations relating the rotation data, where I denote the
+ideal generated by (t − 1) in R.
+Following the method of [22] we lift the equation to Z[t], compute the Taylor expan-
+sions about t = 1 and reduce the coefficients modulo p. Since the indeterminacy of the
+coefficients are determined from the expansion of the cyclotomic polynomial Φp(t) (for
+which p divides the coefficients of its Taylor expansion about t = 1 up to order p − 1) we
+obtain valid congruence relations by equating coefficients modulo p up to order p − 1.
+The expansion arising from contributions from isolated fixed points are given by
+(ta + 1)
+(ta − 1)
+(tb + 1)
+(tb − 1)(t − 1)2 = 4
+ab + 4
+ab(t − 1) + 1
+3
+�a2 + b2 + 1
+ab
+�
+(t − 1)2
+−
+1
+180
+�a4 + b4 − 5a2b2 + 3
+ab
+�
+(t − 1)4 + · · ·
+Similarly the expansion of the second term is given by expressions of the form
+−4αtc
+(tc − 1)2(t − 1)2 = −4α
+c2
++ −4α
+c2 (t − 1) + 1
+3
+α(c2 − 1)
+c2
+(t − 1)2
+− 1
+60
+α(c − 1)(1 + c + c2 + c3)
+c2
+(t − 1)4 + · · · .
+Equating both sides of the expansion (mod p) from the resulting equation in R we thus
+obtain the following congruence relations:
+Theorem 4.1. [22, p. 625] Let (X, π) denote a simply connected, closed 4-manifold with
+a homologically trivial, locally-linear group action of a finite cyclic group π = Z/p of odd
+
+6
+NIMA ANVARI AND IAN HAMBLETON
+order. Then the following congruence relations hold
+(1)
+�
+i
+1
+aibi
+−
+�
+j
+αj
+c2
+j
+≡ 0
+(mod p)
+(2)
+�
+i
+a2
+i + b2
+i
+aibi
++
+�
+j
+αj ≡ 3 Sign(X)
+(mod p)
+(3)
+�
+i
+a4
+i + b4
+i − 5a2
+i b2
+i
+aibi
++ 3
+�
+j
+αjc2
+j ≡ 0
+(mod p)
+(4)
+�
+i
+2a6
+i − 7a4
+i b2
+i − 7a2
+i b4
+i + 2b6
+i
+aibi
++ 10
+�
+j
+αjc4
+j ≡ 0
+(mod p)
+Higher-order relations are valid up to and including terms of order p − 1.
+Example 4.2 (Linear models on CP 2). Let G = Z/p with odd prime p act linearly
+on X = CP 2 by t · [z1 : z2 : z3] = [ζaz1 : z2 : z3] for 0 < a < p.
+The fixed point
+set consists of one isolated fixed point [1 : 0 : 0] with tangential rotation number (a, a)
+and a fixed projective line F = {[z1 : z2 : z3] | z1 = 0} with self-intersection +1 and
+a rotation of cF ≡ a (mod p) on the normal bundle.
+Then it is easy to check that
+the congruence relations are satisfied. Similarly, in the case when the action is given by
+t·[z1 : z2 : z3] = [ζaz1 : ζbz2 : z3] for 0 < a < b < p the action consists of three isolated fixed
+points [0 : 0 : 1], [1 : 0 : 0], [0 : 1 : 0] with rotation numbers (a, b), (b − a, −a), (a − b, −b)
+and with some algebra it can be checked that the congruence relations hold. Additional
+examples can be obtained for #nCP 2 by equivariant connected sums along fixed point
+sets using the linear models.
+5. Equivariant Line Bundles
+Let (X, π) denote a simply connected, closed 4-manifold with a homologically trivial
+action of a finite cyclic group π = Z/p of odd prime p and L → X an equivariant line
+bundle. We compute the contribution of a fixed surface F to the twisted G-signature
+formula:
+L(g, D) | F = {2 coth(iθ/2) − csch2(iθ/2)y}{chg(i∗L)}[F]
+= {2 coth(iθ/2) − csch2(iθ/2)y}{ez+iφ}[F]
+= {2 coth(iθ/2) − csch2(iθ/2)y}{eiφ + eiφz}[F]
+= {2 coth(iθ/2)eiφz − csch2(iθ/2)yeiφ
+= 2c1(i∗L)[F](tcF + 1)
+(tcF − 1)tλ + −4[F]2tcF
+(tcF − 1)2 tλ
+where φ = 2πλ
+p
+and z = c1(i∗L). Similarly for the contribution to the isolated fixed points.
+To summarize, given an equivariant line bundle (L, π) → (X, π), the index of the twisted
+G-signature operator gives a virtual character given by
+
+FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES
+7
+χ(t) =
+�
+i
+(tai + 1)
+(tai − 1)
+(tbi + 1)
+(tbi − 1)tλi +
+�
+j
+−4αjtcj
+(tcj − 1)2tλj +
+�
+j
+2c1(i∗L)[Fj](tcj + 1)
+(tcj − 1)tλj
+where {Fj} are fixed 2-spheres of the action on X with αj denoting the self-intersection
+[Fj] · [Fj]. Note that
+χ(1) = ch(L)L(X)[X] =
+�
+1 + c1(L) + 1
+2c1(L)2
+� �
+4 + p1
+3
+�
+[X]
+(5.1)
+=
+�p1
+3 + 2c1(L)2�
+[X] = Sign(X) + 2c1(L)2[X].
+(5.2)
+Since χ(t) is a virtual character for G = Z/p we may write χ(t) = �p−1
+i=0 aiti for some
+ai ∈ Z and
+χ(t)(t − 1)2 = χ(1)(t − 1)2 + higher order terms
+(mod p)
+We can then take the Taylor expansion of the right-hand side of the twisted G-signature
+formula after multiplying by (t−1)2 and equate the first and second order terms to obtain
+two additional congruence relations. The first order term vanishes while the second order
+term is congruent to χ(1) (mod p).
+Taking Taylor expansions for these terms in the
+G-signature formula give:
+(ta + 1)
+(ta − 1)
+(tb + 1)
+(tb − 1)(t − 1)2tλ = 4
+ab + 4(λ + 1)
+ab
+(t − 1)+
+1
+3
+(a2 + b2 + 1 + 6λ2 + 6λ)
+ab
+(t − 1)2 + · · · .
+Similarly for the second term:
+−4αtc
+(tc − 1)2(t − 1)2tλ = −4α
+c2
++ −4α(1 + λ)
+c2
+(t − 1)+
+1
+3
+α(c2 − 1 − 6λ − 6λ2)
+c2
+(t − 1)2 + · · · .
+and for the third term:
+2m(tc + 1)
+(tc − 1)(t − 1)2tλ = 4m
+c (t − 1) + 2m(2λ + 1)
+c
+(t − 1)2 + · · ·
+where m = c1(i∗L)[F]. Combining these expressions we obtain the following theorem:
+Theorem 5.1. Let (L, π) → (X, π) denote an equivariant line bundle over a simply
+connected, closed 4-manifold with a homologically trivial group action of a finite cyclic
+
+8
+NIMA ANVARI AND IAN HAMBLETON
+group π = Z/p of odd order. Then the following congruence relation holds
+(i)
+�
+i
+λi
+aibi
+−
+�
+j
+λjαj
+c2
+j
++
+�
+j
+c1(i∗L)[Fj]
+cj
+≡ 0
+(mod p)
+(ii)
+�
+i
+λ2
+i
+aibi
+−
+�
+j
+λ2
+jαj
+c2
+j
++ 2
+�
+j
+λjc1(i∗L)[Fj]
+cj
+≡ c1(L)2[X]
+(mod p).
+Example 5.2. Let π = Z/p act on X = CP 2 preserving an almost complex structure.
+Then the complexified second exterior power of the tangent bundle is an equivariant line
+bundle see [22, Proposition 1.8]).
+Example 5.3 (Linear models on CP 2). Let p denote an odd prime and consider X = CP 2
+and a linear action t·[z1 : z2 : z3] = [ζaz1 : ζbz2 : z3] for 0 < a < b < p. We give an explicit
+construction of equivariant line bundles over X. Consider a finite dimensional complex
+representation space V = C(λ1) ⊕ C(λ2) ⊕ C(λ3) with action given by ρ ∈ GL3(C):
+(5.4)
+ρ =
+
+
+tλ1
+tλ2
+tλ3
+
+ : C3 \ {0} −→ C3 \ {0}
+with the λi positive integer weights. Let S(V ) denote the unit sphere in V then ρ com-
+mutes with the free S1-action on S(V ) and
+CP 2 = S(V )/S1 = S5/S1.
+The ρ-action on S(V ) is a lift of the linear action on X if the following system of linear
+congruences
+λ1 − λ3 ≡ a,
+λ2 − λ3 ≡ b
+λ2 − λ1 ≡ b − a,
+λ3 − λ1 ≡ −a
+λ1 − λ2 ≡ a − b,
+λ3 − λ2 ≡ −b.
+has a solution. This system has one degree of freedom; let λ3 = λ be a fixed parameter,
+then the isotropy representations over the three isolated fixed points are given by:
+fixed point p1 = [0 : 0 : 1], rotation number (a, b), with isotropyλ3 ≡ λ
+fixed point p2 = [1 : 0 : 0], rotation number (b − a, −a), with isotropyλ1 ≡ λ + a
+fixed point p3 = [0 : 1 : 0], rotation number (−b, a − b)with isotropyλ2 ≡ λ + b.
+The equivariant line bundle
+L = S(V ) ×S1 C
+is the canonical bundle over CP 2 and the congruence relations of the theorem are satisfied:
+�
+i
+λi
+aibi
+≡ 0
+�
+i
+λ2
+i
+aibi
+≡ 1.
+
+FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES
+9
+In the case when b = 0 the action is given by t · [z1 : z2 : z3] = [ζaz1 : z2 : z3] and has
+a fixed projective line F = {z1 = 0} with normal rotation number cF ≡ a (mod p) and
+self-intersection +1, while the isolated fixed point [1 : 0 : 0] has rotation number (a, a).
+The compatibility for a lift of the linear action is the congruence relation:
+λ − λF ≡ a
+(mod p).
+It is easily seen that the congruence relation of the theorem are satisfied:
+λ
+a2 − λF
+a2 + c1(i∗L)[F]
+a
+≡ 0.
+where we used c1(i∗L)[F] = −1 since the first Chern class of the canonical line bundle
+over CP 2 is negative of the preferred generator [F] ∈ H2(CP 2; Z) [30, Theorem 14.10,
+p. 169].
+Similarly, the second relation is
+λ2
+a2 − λ2
+F
+a2 + 2λFc1(i∗L)[F]
+a
+≡ (a + λF)2 − λ2
+F
+a2
++ 2λFc1(i∗L)[F]
+a
+≡ 1 ≡ c1(L)2[X].
+Definition 5.5. Let (X, π) be a homologically trivial of π = Z/p in the setting of Theorem
+A, with rotation data F = {(ai, bi), (cj, αj) | i ∈ I, j ∈ J}. We say that (X, π) satisfies the
+condition of Theorem A if there exists a set of isotropy data I = {λi, λj | i ∈ I, j ∈ J},
+and a set of integers {mj | j ∈ J} so that the equation given in Theorem A holds.
+In the next statement, we apply the equivariant connected sum operation to line bun-
+dles.
+Lemma 5.6. Suppose that (X, π) satisfies the condition of Theorem A. There there is an
+equivariant connected sum (X♯ CP 2, π) with a linear π-action on CP 2 which also satisfies
+the condition of Theorem A.
+Proof. We first suppose that (X, π) contains a fixed 2-sphere Fj with data {αj, cj}. Let
+(CP 2, π) be the linear action given by t · [z1 : z2 : z3] = [ζ−cjz1 : z2 : z3] as in Example 4.2.
+We do the equivariant connected sum (preserving the orientations) along a point in Fj
+and a point in the fixed 2-sphere of CP 2. The new data is obtained by (i) adding the data
+{(−cj, −cj); λj} for the newly created isolated fixed point (on CP 2), and (ii) the data
+{(cj, αj + 1); λj} for the new fixed 2-sphere. With these choices, it follows that the action
+(X♯ CP 2, π) satisfies the condition of Theorem A. The proof in case the action (X, π) has
+only isolated fixed points is easier, and will be left to the reader.
+□
+Remark 5.7. Suppose that (X, π) has data satisfying the condition of Theorem A, and
+contains a fixed 2-sphere F. Let X0 ⊂ X denote the complement of a linear π-invariant
+4-ball neighbourhood of a point x ∈ F. If L is an equivariant line bundle over (X♯ CP 2, π),
+then the restriction of L to X0 extends to an equivariant line bundle over (X, π) realizing
+the given data.
+
+10
+NIMA ANVARI AND IAN HAMBLETON
+6. The proof of Theorem A
+The first relation in Theorem 5.1 proves the necessary conditions of Theorem A. To
+prove sufficiency we will need the following lemmas.
+Note that in a standard lens space Y = L(n; a, b), a generator µ ∈ H1(Y ; Z) is repre-
+sented by a circle fibre in the fibration S1 → L(n; a, b) → S2 given by the quotient of a
+free Z/n action on S3.
+Lemma 6.1. The linking paring lk: H1(Y ) × H1(Y ) → Q/Z in the lens space Y =
+L(n; a, b) is given by lk(µ, µ) = ab
+p where µ is a generator of H1(Y ; Z) = Z/n.
+Proof. In the usual representation of lens spaces L(n; q) as the quotient of the free Z/n
+action t · (z1, z2) = (ζz1, ζqz2) on S3, the linking pairing is given by lk(µ, µ) = q
+n. The
+diffeomorphism L(n; a∗b) → L(n; a, b) arising from changing the generator in Z/n induces
+a map on first homology given by multiplication by a. It follows that the linking pairing
+on Y is given by
+lk(aµ, aµ) = a2 ·
+�a∗b
+n
+�
+= ab
+n ∈ Q/Z.
+□
+Lemma 6.2 ((See [22, Proposition 1.4, p. 621], [25, Lemma 2.11, p. 95])). Let φ: π1(Y ) −→
+U(1) be the holonomy representation of a flat U(1)-bundle over the lens space Y
+=
+L(n; a, b) that sends a generator µ to exp(2πiλ/n). Then the Poincar´e dual of the first
+Chern class PD(c1(L)) is given by λ
+ab[µ] in H1(Y ; Z).
+Proof. The adjoint to the linking form Φ: H1(Y ; Z) → Hom(H1(Y ; Z), Q/Z) sends m[µ]
+to lk(mµ, −) which can be identified with the holonomy representation of the flat bundle
+via :
+e2πi·lk(mµ,−) : H1(Y ; Z) → U(1)
+and this maps the generator to exp(2πi · mab/n). It follows that m ≡ λ
+ab (mod n).
+□
+If Y is a lens space, we let ˆµ ∈ H2(Y ; Z) denote a standard cohomology generator: the
+Poincar´e dual to the circle fibre class µ ∈ H1(Y ; Z).
+Lemma 6.3. Let u: Y ′ → Y be a d-fold regular covering of lens space, where H1(Y ; Z) ∼=
+Z/dn and H1(Y ′; Z) ∼= Z/n, with gcd(d, n) = 1. Then u∗(ˆµ) = ˆµ′ ∈ H2(Y ′; Z).
+Proof. For a d-fold regular covering u: Y ′ → Y of lens spaces, we have u∗[µ′] = d[µ] ∈
+H1(Y, Z) and u∗[Y ′] = d[Y ] ∈ H3(Y ; Z).
+The cohomology generator ˆµ = [Y ] ∩ µ ∈
+H2(Y ; Z) is the Poincar´e dual of µ, and similarly for ˆµ′ ∈ H2(Y ′; Z). We have the formula
+u∗([Y ′] ∩ u∗(ˆµ)) = u∗(µ′) = dµ.
+If H1(Y ; Z) = Z/dn and H1(Y ′; Z) = Z/n, where gcd(d, n) = 1, then u∗(ˆµ) = kˆµ′ implies
+that k ≡ 1 (mod n). Hence u∗(ˆµ) = ˆµ′ ∈ H2(Y ′; Z).
+□
+
+FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES
+11
+Suppose that (X, π) satisfies the assumptions of Theorem 5.1. Let Σ ⊂ X denote the
+singular set of the action, and let X0 := X − Σ. Write αj = pajαj, where αj is prime to
+p (for each fixed 2-sphere Fj).
+Lemma 6.4. If the singular set Σ ⊂ X contains an isolated point, then H1(X0; Z) is a
+quotient of � Z/αj and has order prime to p.
+Proof. Let F ⊂ X denote the disjoint union of the fixed 2-spheres, so F = �
+j Fj. First
+note that H1(X0) ∼= H3(X, F) and we have an exact sequence
+· · · → H2(X) → H2(F) → H3(X, F) → H3(X) → . . .
+Since H3(X) = 0, and the homology classes of the fixed 2-spheres are linearly independent
+mod p (by [9, Corollary 2.6]), it follows that H3(X, F) ∼= H1(X0) is a torsion group of
+order prime to p.
+Moreover, the exact Mayer-Vietoris sequence
+0 → H2(X0) ⊕ H2(F) → H2(X) → H1(∂X0) → H1(X0) → 0
+and the equality H1(∂X0) = H1(∂ν(F)) ∼= � Z/αj completes the proof.
+□
+The proof of Theorem A. By Theorem 5.1(i), the indicated formulas hold if (X, π) admits
+an equivariant line bundle. It remains to prove that a solution {λi, λj, mj | i ∈ I, j ∈ J}
+to the congruence relation
+�
+i
+λi
+aibi
+−
+�
+j
+λjαj
+c2
+j
++
+�
+j
+mj
+cj
+≡ 0
+(mod p)
+is sufficient for the existence of an equivariant line bundle with {λi, λj} isotropy repre-
+sentations over the isolated fixed points and 2-spheres respectively, and mj ≡ c1(i∗L | Fj)
+mod αj. For simplicity, we will assume that the action (X, π) contains at least one iso-
+lated fixed point. This may always be arranged by taking the equivariant connected sum
+of (X, π) along a fixed 2-sphere with a suitable linear π-action on CP 2 (see Lemma 5.6).
+Let X0 = X−N, where N = ν(Σ) is a π-invariant tubular neighbourhood of the singular
+set Σ ⊂ X. More explicitly, X0 is the compact 4-manifold with boundary obtained by
+removing π-invariant 4-balls around each isolated fixed point and π-invariant tubular
+neighbourhoods D2 → ν(Fj) → Fj around each π-fixed 2-sphere Fj with rotation tcj on
+D2-fibers. Then ν(Fj) is a 2-disk bundle over S2 with Euler class αj[F] ∈ H2(F; Z) ∼= Z,
+and the lens space ∂ν(Fj) = L(αj, 1) inherits a free Z/p action with rotation number cj
+on the circle fibre.
+If W0 := X0/π denotes the quotient manifold with (regular) covering map q: X0 → W0
+classified by u: W0 → Bπ, then the boundary ∂W0 consists of lens spaces Yi = L(p; ai, bi)
+and Yj = L(pαj; cj, cj). Note that
+H1(W0; Z) ∼= Z/p ⊕ H1(X0, Z)
+by the spectral sequence of the covering.
+By Lemma 6.4, H1(X0; Z) is a quotient of
+� Z/αj and has order prime to p.
+Recall that π-equivariant line bundles L over (X, π) are classified by an element
+θ(L) ∈ H2
+π(X; Z) = H2(X ×π Eπ; Z)
+
+12
+NIMA ANVARI AND IAN HAMBLETON
+in the Borel requivariant cohomology of X (see [27]). Since the π-action on X0 is free, for
+the restriction L0 ց X0 we have θ(L0) ∈ H2
+π(X0; Z) ∼= H2(W0; Z), and θ(L0) = c1(¯L0),
+where ¯L0 is the line bundle over W0 obtained by dividing out the free π-action on the
+total space of L0. Moreover, in the short exact sequence
+0 → H2(Z/p; Z)
+c∗
+−→ H2(W0; Z)
+q∗
+−→ H2(X0; Z) → 0
+the pullback q∗(θ(L0)) = c1(q∗(¯L0)) = c1(L0) ∈ H2(X0; Z).
+The strategy will be to find a suitable element θ(L) ∈ H2
+π(X; Z) by studying the Mayer-
+Vietoris sequence
+· · · → H2
+π(X) → H2
+π(X0) ⊕ H2
+π(N) → H2
+π(∂X0)
+δ−→ H3
+π(X) → H3
+π(X0) ⊕ H3
+π(N) → . . .
+in Borel cohomology associated to the π-equivariant decomposition X = X0 ∪ N.
+We observe the following:
+(i) The Mayer-Vietoris coboundary map H2
+π(∂X0)
+δ−→ H3
+π(X) factors
+H2
+π(∂X0)
+δ−→ H3
+π(X0, ∂X0) ∼= H3
+π(X, N) → H3
+π(X).
+(ii) The cokernel of the map H2
+π(N) → H2
+π(∂X0) has exponent p. This follows from
+the commutative diagram of restriction maps
+H2
+π(N)
+�
+�
+H2
+π(∂X0)
+�
+� � Z/pαj
+�
+H2(N)
+� H2(∂X0)
+∼
+= � � Z/αj
+since the map H2
+π(N) → H2(N) is surjective (by the Borel spectral sequence) and
+the map H2(N, ∂N) → H2(N) is adjoint to the (diagonal) intersection form on
+N, with cokernel H2(∂N) = H2(∂X0), hence determined by the self-intersection
+numbers {αj}.
+(iii) We have H3
+π(X0, ∂X0) ∼= H3(W0, ∂W0) ∼= H1(W0) ∼= Z/p⊕H1(X0), where H1(X0)
+is a quotient of � Z/αj and has order prime to p (by Lemma 6.4).
+To complete the proof of Theorem A, it is now enough to produce a class
+θ0 ∈ H2
+π(X0) ∼= H2(W0),
+which added together with the classes already found in H2
+π(N) will have image zero in
+H2
+π(∂X0). By the observations above, this amounts to finding a U(1)-bundle ¯L0 on ∂W0
+whose first Chern class θ0 = c1(¯L0) has image of order prime to p under the coboundary
+map
+H2(∂W0) → H3(W0, ∂W0).
+In other words, we need to find suitable U(1)-bundles over each of boundary components
+of ∂W0, so that the sum of their first Chern classes is zero (mod p). This required relation
+is exactly the condition (1.1) given in the statement of Theorem A.
+Let Y denote one of the lens spaces in ∂W0, For convenience, we will identify Z/p ∼=
+H2(Y ; Z) by a �→ a · ˆµ, for a ∈ Z/p, where ˆµ ∈ H2(Y ; Z) denotes a standard generator,
+Poincar´e dual to the circle fibre class µ ∈ H1(Y ; Z) (introduced in Lemma 6.1).
+
+FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES
+13
+If Y = L(p; ai, bi) then choose the holonomy representation that sends a generator in
+π1(Y ) to exp(2πiλi/p). The first Chern class of the associated flat U(1)-bundle is
+λi
+aibi
+[ˆµ] ∈ H2(Yi; Z) ∼= Z/p.
+In the case when the π-action on X has only isolated fixed points, the condition for an
+extension is that these elements lie in the kernel of δ in H2(∂W0; Z) = �
+i H2(Yi; Z) which
+is equivalent to the condition �
+i
+λi
+aibi
+≡ 0 (mod p).
+In the general case when components of the fixed set contain 2-spheres, we need to
+consider contributions from the lens spaces Yj = L(pαj; cj, cj). These lens spaces arise
+from the free tcj-action on �Yj := ∂ν(Fj) ≈ L(αj; 1). Conisder the induced covering spaces
+of Yj by the lens spaces L(paj+1, 1) and L(αj, 1), where αj = pajαj, and note that the
+covering maps induce an isomorphism:
+(6.5)
+H2(Yj; Z)
+∼
+=
+�
+∼
+=
+�
+H2(L(paj+1, 1)) ⊕ H2(L(αj, 1))
+∼
+=
+�
+Z/pαj
+∼
+=
+� Z/paj+1 ⊕ Z/αj
+Under the two covering maps, the standard cohomology generator ˆµj ∈ H2(Yj; Z) is sent
+to the standard generators in H2(L(paj+1, 1)) and H2(L(αj, 1)), respectively, by Lemma
+6.3. The maps in the lower sequence are the reductions mod paj+1 and α, after using the
+identifications provided by the cohomology generators.
+Now the following congruences uniquely determines a Chern class c1(¯Lj) = ℓj
+c2
+j
+[ˆµj] ∈
+H2(Yj; Z), by choosing:
+(6.6)
+ℓj
+≡
+−λjαj
+(mod paj+1),
+ℓj
+≡
+cjmj
+(mod αj).
+and hence a U(1)-bundle ¯Lj ց Yj. The minus sign is chosen in the first congruence
+because the induced orientation on Yj from ∂W0 is opposite to its orientation as the disk
+bundle over S2 with Euler class αj.
+By diagram (6.5) and Lemma 6.3, the first Chern class has image
+c1(¯Lj) = ℓj
+c2
+j
+[µj] = −λjαj + cjmj
+c2
+j
+[µj] ∈ H2(Yj; Z)
+with respect to the decomposition H2(Yj; Z) ∼= Z/paj+1 ⊕ Z/αj. After substituting these
+expressions into the formula of Theorem 5.1, we see that the sum vanishes mod p, and
+hence the required line bundle ¯L0 over W0 exists.
+□
+7. Equivariant SU(2) Bundles
+In this section we compute a (necessary) congruence relation similar to the previous
+section, but for equivariant SU(2)-bundles.
+As above, we work over a closed, simply
+connected, oriented 4-manifold with a finite homologically trivial cyclic group action. We
+
+14
+NIMA ANVARI AND IAN HAMBLETON
+again use the twisted G-signature formula (with the previously established notation). In
+particular, let D denote the signature operator twisted by an equivaraint SU(2)-bundle
+E −→ X, then the contribution to the Lefschetz numbers from isolated fixed points is
+given by
+L(g, D) | pt = (ta + 1)
+(ta − 1)
+(tb + 1)
+(tb − 1)(tλ + t−λ).
+We need to compute the contribution from isolated fixed 2-spheres F.
+Since E | F =
+L ⊕ L−1, we have chg(L ⊕ L−1 | F) = {eλ+z + e−λ−z}[F] and
+L(g, D) | F = {2 cot(iθ/2) − csch2(iθ/2)y} chg(L ⊕ L−1 | F)[F]
+= {2(tc + 1)
+(tc − 1) −
+4tcy
+(tc − 1)2}{eλ(1 + z) + e−λ(1 − z)}[F]
+= {2(tc + 1)
+(tc − 1) −
+4tcy
+(tc − 1)2}{tλ + t−λ + z(tλ − t−λ)}[F]
+= − 4tc[F]2
+(tc − 1)2(tλ + t−λ) + 2c1(L)[F](tc + 1)
+(tc − 1)(tλ − t−λ).
+Also note
+χ(1) = ch(E)L(X)[X] = (2 − c2(E))(4 + 1
+3p1)
+= 2 Sign(X) − 4c2(E).
+We now again multiply both sides of the G-signature formula by (t − 1), take Taylor
+expansions about t = 1 and reduce coefficients modulo p:
+(ta + 1)
+(ta − 1)
+(tb + 1)
+(tb − 1)(t − 1)2(tλ + t−λ) = 8
+ab + 8
+ab(t − 1)+
+2
+3
+(a2 + b2 + 1 + 6λ2)
+ab
+(t − 1)2 + · · · .
+and for the second term, where we let m denote c1(L)[F]:
+(t − 1)2{ −4αtc
+(tc − 1)2(tλ + t−λ) + 2m(tc + 1)
+(tc − 1)(tλ + t−λ)}
+= −8α
+c2
++ −8α
+c2 (t − 1) + 2
+3
+(αc2 − α − 6αλ2 + 12mcλ)
+c2
+(t − 1)2 + · · ·
+Summing over all the fixed sets and simplifying the coefficient of second order term (t−1)2,
+we obtain:
+2 Sign(X) +
+�
+i
+4λ2
+i
+aibi
+−
+�
+j
+4αjλ2
+j
+c2
+j
++
+�
+j
+8mjλj
+cj
+.
+Equating this with χ(1) = 2 Sign(X) − 4c2(E) and reducing coefficients modulo p gives
+the following congruence relation:
+
+FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES
+15
+Theorem 7.1. Let (E, π) → (X, π) denote an equivariant SU(2)-bundle over a simply
+connected, closed 4-manifold with a homologically trivial group action of a finite cyclic
+group π = Z/p of odd prime order. Then the following congruence relation holds
+�
+i
+λ2
+i
+aibi
+−
+�
+j
+αjλ2
+j
+c2
+j
++
+�
+j
+2λj
+cj
+c1(i∗Lj)[Fj] ≡ −c2(E)[X]
+(mod p),
+where Lj is a local reduction E | Fj = Lj ⊕ L−1
+j .
+Example 7.2 (Linear Models on S4). Let X = S4 with a linear Z/p-action which gives
+rotation numbers (a, b) and (a, −b). Let E denote the instanton one equivariant SU(2)-
+bundle, i.e. with c2(E) = 1. Then the congruence relation is given by
+−c2(E) = λ2
+1
+ab − λ2
+2
+ab
+(mod p)
+It is elementary to check that for congruence relation is satisfied with the following isotropy
+representations
+λ1 = b − a
+2
+λ2 = a + b
+2
+.
+over the fibres of the fixed points.
+Example 7.3 (Linear Models on CP
+2). Let X = CP
+2 with a linear Z/p-action with
+one isolated fixed point with rotation number (a, −a) for some a (mod p) and a fixed
+projective line F with rotation number a (mod p) on the normal bundle. Let E again
+denote the instanton one equivariant SU(2)-bundle. The congruence relation gives
+−1 ≡ −λ2
+a2 + λ2
+F
+a2 + 2mλF
+a
+where m = c1(i∗L)[F]. There exists two distinct lifts giving rise to equivariant bundles
+which admit G-invariant ASD connections. In the case when the equivariant lift comes
+from ”bubbling” on the isolated fixed point then m = 0 and
+λ ≡ a
+(mod p)
+λF ≡ 0
+(mod p).
+Thus the congruence is satisfied. On the other hand, if we choose the equivariant lift
+associated to the fixed 2-sphere (from 3-dimensional fixed connected component in the
+moduli space of equivariant ASD connections with c2(E) = 1) then m = −1 and
+λ ≡ a/2
+(mod p)
+λF ≡ a/2
+(mod p),
+again the congruence relation is satisfied.
+In the next section we compute the dimension of the moduli space of invariant anti-self
+dual connections for a given equivariant SU(2)-bundle.
+
+16
+NIMA ANVARI AND IAN HAMBLETON
+8. Equivariant Index Computation
+Let X be a simply connected, closed, smooth negative definite 4-manifold, with a
+homologically trivial action of a finite group G.
+If E ց X is an SU(2)-bundle with
+c2(E) = k, the moduli space M∗
+1(X) of irreducible ASD connections (on an SU(2)-bundle
+E with c2(E) = 1) inherits a G-action, and the connected components of the fixed point
+set MG
+1 (X) correspond to G-invariant ASD connections for certain equivariant lifts of the
+G-action on X to E (see [15], [6], [20, Theorem A], [24, §2]).
+We want to compute the dimension of the moduli space MG
+k (X) of irreducible G-
+invariant ASD connections. This is motivated by the following example, for which the
+formal dimension dim M∗
+1(X) = 5.
+In this case, we expect a dimension formula that
+gives 1 and 3-dimensional strata depending on contributions from isolated fixed points
+or isolated fixed 2-spheres in X and on the isotropy representations from the equivariant
+lift (see [6] and [21] for details). There are similar index calculations in the literature in
+various gauge-theoretic settings (for example, see [13, §3], [4]), [28, 29], [35], [1]).
+We first very briefly review the dimension calculation in the non-equivariant setting to
+set some notation. Let D+
+A = d∗
+A + d+
+A : Ω1(ad E) → Ω0(ad E) ⊕ Ω2
++(ad E) denote the
+anti-self duality operator, and let Mk denote the ASD moduli space with c2(E) = k. Note
+that the formal dimension is given by dim Mk = − Ind(D+
+A) and this is given by
+(8.1)
+Ind(D+
+A) = ˆA(X) ch(S+) ch(adC E)[X]
+where S = S+ ⊕S− and ˆA(X) = �
+xi/2
+sinh(xi/2) with ch(S) = �(exi/2 + e−xi/2) and ch(S+) −
+ch(S−) = �(exi/2 − e−xi/2). Using this we compute
+2 ˆA(X) ch(S+) ch(adC E)[X] = (4 + 1
+3p1 + χ)(3 − 4c2(E))[X]
+= −16c2(E) + 3(p1
+3 + χ).
+Thus the index Ind(D+
+A) = −8c2(E) + 3
+2(Sign +χ)(X) and we get the usual expression
+dim Mk = 8k − 3/2(χ + Sign)(X) for the dimension of the moduli space. Also note the
+following alternative expression for the index:
+Ind(D+
+A) = ch(S+ − S−) ch(S+) ch(adC E)Td(TX ⊗ C)
+e(X)
+[X]
+= ˆA(X) ch(S+ ⊗ adC E)[X].
+For the equivariant setting E is an equivariant SU(2)-bundle and let D = D+
+A denote the
+anti-self duality operator d∗
+A + d+
+A : Ω1(ad E)G → Ω0(ad E)G ⊕ Ω2
++(ad E)G. We compute
+the equivariant index by averaging the Lefschetz numbers as in [13]:
+Ind(D) = 1
+p
+�
+g∈G
+L(g, D)
+
+FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES
+17
+Ind(D) = 1
+p{L(1, D) +
+�
+g̸=1
+L(g, D)}
+= 1
+p{−8c2(E) + 3
+2(χ + Sign)(X) +
+�
+g̸=1
+L(g, D)}
+= 1
+p{−8c2(E) + 3p
+2 (χ + Sign)(X/G) − 3
+2(dχ + dσ)(XG) +
+�
+g̸=1
+L(g, D)}
+where pχ(X/G) = χ(X) + dχ with dχ = �
+g̸=1 χ(Xg) is the Euler characteristic defect
+terms and similarly for the signature defect term:
+− 3
+2(dχ + dσ)[pt] = −3
+2(1 − cot(θ1/2) cot(θ2/2))
+− 3
+2(dχ + dσ)[F] = −3
+2(2 + [F]2 csc2(θ/2)),
+where (θ1, θ2) are the rotation numbers at an isolated fixed point and θ = cF is the rotation
+number on the normal bundle to F. Decomposing the contributions from isolated fixed
+points and 2-spheres:
+�
+g̸=1
+L(g, D)(XG) =
+�
+g̸=1
+{
+�
+i
+L(g, D) | (ai,bi) +
+�
+j
+L(g, D) | Fj}.
+Now chg(adC E)(pt) = 3 − 4 sin2( πkℓ
+p ), with ℓ the isotropy representation on the fiber of E
+over the fixed point. The Lefshetz numbers from the fixed sets can be computed directly
+from the index formula and are given by:
+(8.3)
+L(g, D) | pt = −1
+2 [cot(θ1/2) cot(θ2/2) − 1] chg(adC E)[pt]
+(8.4)
+L(g, D) | F = [−i cot(θ/2) + 1
+2(χ + csc2(θ/2)y)] chg(adC E)[F],
+with χ the Euler class of the tangent bundle to F and y is the Euler class of the normal
+bundle to F. We first compute the contribution from isolated fixed points.
+L(g, D) | pt = −1
+2 [cot(θ1/2) cot(θ2/2) − 1][3 − 4 sin2(πkℓ
+p )][pt]
+= −3
+2[cot(θ1/2) cot(θ2/2) − 1] − 2 sin2(πkℓ
+p )
++ 2 cot(θ1/2) cot(θ2/2) sin2(πkℓ
+p ).
+
+18
+NIMA ANVARI AND IAN HAMBLETON
+Summing over all isolated fixed points gives
+1
+p
+�
+g̸=1
+�
+i
+L(g, D) | (ai,bi) = 3
+2p
+�
+i
+(dχ + dσ)(ai, bi) − 2
+p
+�
+i
+p−1
+�
+k=1
+sin(πkℓi
+p )
++ 2
+p
+�
+i
+p−1
+�
+k=1
+cot(aiπk
+p
+) cot(biπk
+p ) sin2(πkℓi
+p )
+= 3
+2p
+�
+i
+(dχ + dσ)(ai, bi) + m +
+�
+i
+ρL(p, ai, bi, ℓi)
+where m is the number isolated fixed points with non-trivial representation on the fiber
+and ρL(p, a, b, ℓ) is the rho invariant of lens spaces.
+We need to compute chg(adC E | F). Since an SU(2) bundle restricted over a fixed 2-
+submanifold has a local abelian reduction E | F = L ⊕ L−1 for some L, we have ad E | F =
+L2 ⊕ R. We need to compute chg(adC E | F) = chg(L2) + chg(L2) + 1 and this contributes
+chg(adC E | F) = (g + gc1(L2)) + (g−1 + g−1c1(L2)) + 1
+= (g + g−1 + 1) + c1(L2)(g − g−1)
+= (3 − 4 sin2(πkℓ
+p )) + 2ic1(L2) sin(2πkℓ
+p
+),
+where now ℓ is the isotropy representation on the fibre over the fixed 2-sphere F. Substi-
+tuting these terms, the Lesfchetz number L(g, D) | F evaluated on fixed 2-spheres gives:
+L(g, D) | F = [−i cot(θ/2) + 1
+2(χ + csc2(θ/2)y)][(3 − 4 sin2(πkℓ
+p )) + 2ic1(L2) sin(2πkℓ
+p
+)][F]
+= 1
+2[χ + csc2(θ/2)y][3 − 4 sin2(πkℓ
+p )] + 2c1(L2) sin(2πkℓ
+p
+) cot(θ
+2).
+Let us introduce a kind of rho invariant term for fixed surfaces:
+ρF(ℓ) = 2
+p
+p−1
+�
+k=1
+csc2(πcFk
+p
+) sin2(πkℓ
+p )[F]2 − 4c1(L)[F]
+p
+p−1
+�
+k=1
+sin(2πkℓ
+p
+) cot(πkcF
+p
+),
+with this notation we have
+1
+p
+�
+g̸=1
+�
+j
+L(g, D) | Fj = 3
+2p
+�
+j
+(dχ + dσ)[Fj] − 2
+p
+�
+j
+χ(Fj)
+p−1
+�
+k=1
+sin2(πkℓj
+p
+)
+−
+�
+j
+ρFj(ℓj).
+
+FINITE GROUP ACTIONS ON 4-MANIFOLDS AND EQUIVARIANT BUNDLES
+19
+Now combining all the terms we obtain:
+Ind(DA) = −8
+p c2(E) + 3
+2(χ + Sign)(X/G)) − m +
+�
+i
+ρL(p, ai, bi, ℓi)
+−
+�
+j with ℓj̸=0
+χ(Fj) −
+�
+j
+ρFj(ℓj).
+Since dim MG
+k (X) = − Ind(DA), the dimension formula is
+dim MG
+k (X) = 8
+pc2(E) − 3
+2(χ + Sign)(X/G) + m −
+�
+i
+ρL(p, ai, bi, ℓi)
++
+�
+j with ℓj̸=0
+χ(Fj) +
+�
+j
+ρFj(ℓj).
+Before giving an example we note a few special cases.
+When the action on X only
+has isolated fixed points, let (ai, bi) denote the rotation numbers and ℓi the isotropy
+representation over the points, the formula reduces to the following:
+dim MG
+k (X) = 8c2(E)
+p
+− 3
+2(χ + Sign)(X/G) + m −
+�
+i
+ρL(p, ai, bi, ℓi).
+For invariant ASD connections on the four-sphere this formula reduces to that of [4, p.
+394]. In the case of SO(3)-bundles in the orbifold setting, see Fintushel and Stern [13].
+When the action on X is a smooth involution with fixed 2-sphere and non-trivial action
+on fibre cF ≡ ℓ ≡ 1 mod 2 the formula above reduces to:
+dim MG
+k (X) = 4c2(E) − 3
+2(χ + Sign)(X/G) + χ(F) + [F]2.
+which matches with Wang [35, Theorem 18, p. 130].
+We finish this section with an
+example.
+Example 8.5. Let X = #3CP
+2 with a linear Z/p-action with p = 5 that arises from
+equivariant connected sums of linear actions in the following way. Take the equivariant
+connected sum of two copies of CP
+2 along the two dimensional fixed sets which fixes a pro-
+jective line and a rotation number of (1, −1) at the isolated fixed points in each copy. Now
+at one of the isolated fixed points take the equivariant connected sum with CP
+2 that has
+a linear action with 3 isolated fixed points with rotation numbers (1, 1), (2, −1), (2, −1).
+The result is a smooth, homologically trivial Z/5-action on X that has 3 isolated fixed
+points with rotation data {(1, −1), (2, −1), (2, −1)} and a single fixed 2-sphere F with
+rotation number cF ≡ 1 (mod p) on the normal bundle and has self intersection −2.
+The compactified, equivariant ASD instanton one moduli space M1(X) has dimension
+5 with fixed components that are 1 and 3-dimensional which correspond to invariant ASD
+connections for a lifted action to the SU(2)-bundle (see [21]).
+The boundary of the moduli space is the ”bubbling” of highly concentrated ASD con-
+nections which can be identified with a copy of X. The isolated fixed points propagate
+
+20
+NIMA ANVARI AND IAN HAMBLETON
+1-fixed dimensional strata into the moduli space. We will compute the dimension of these
+strata using the dimension formula from this section and from the fixed point data.
+For example, at the isolated fixed point (2, −1) the highly concentrated instantons
+correspond to ASD connections on the 4-sphere, with equivariant lifts matching the linear
+models which then pull back to X using the degree 1-map in the formation of the Taubes
+boundary. This determines the equivariant lift on X and has isotropy representation tλ1
+over the fixed point (2, −1) with λ1 ≡ −3 (mod p) and tλ2 over all the other fixed point
+sets with λ2 ≡ 1 (mod p). The dimension formula gives:
+8
+p − ρL(p, 2, −1, −3) − ρL(p, 2, −1, 1) − ρL(p, 1, −1, 1) + χ(F) + ρF(1) = 1.
+On the other hand, at a point on the fixed 2-sphere F following the same procedure with
+the degree one Taubes map, we can pull-back an equivariant bundle from the linear model
+on S4 with a fixed embedded 2-sphere. This time we get an equivariant SU(2)-bundle on
+X with c1(L)[F] = −1 in the local reduction E | F = L⊕L−1. The isotropy representation
+is tλ over all the fixed point sets with λ ≡ 1 (mod p). We then have:
+8
+p − 2ρL(p, 2, −1, 1) − ρL(p, 1, −1, 1) + χ(F) + ρF(1) = 3.
+after substituting the data into the dimension formula.
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+Department of Mathematics & Statistics, McMaster University L8S 4K1, Hamilton,
+Ontario, Canada
+Email address: anvarin@math.mcmaster.ca
+Department of Mathematics & Statistics, McMaster University L8S 4K1, Hamilton,
+Ontario, Canada
+Email address: hambleton@mcmaster.ca
+

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+Notice: This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-
+AC0500OR22725 with the U.S. Department of Energy. The United States Government retains 
+and the publisher, by accepting the article for publication, acknowledges that the United States 
+Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or 
+reproduce the published form of this manuscript, or allow others to do so, for the United States 
+Government purposes. The Department of Energy will provide public access to these results of 
+federally sponsored research in accordance with the DOE Public Access Plan 
+(http://energy.gov/downloads/doe-public-access-plan). 
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+Discovery of structure-property relations for molecules via hypothesis-driven active 
+learning over the chemical space 
+ 
+Ayana Ghosh,1 Sergei V. Kalinin2 and Maxim A. Ziatdinov1,3 
+ 
+1 Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, 
+TN 37831 USA 
+2Department of Materials Science and Engineering, University of Knoxville, Knoxville, TN 
+37996 USA  
+3Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 
+37831 USA 
+ 
+ 
+Discovery of the molecular candidates for applications in drug targets, biomolecular systems, 
+catalysts, photovoltaics, organic electronics, and batteries, necessitates development of machine 
+learning algorithms capable of rapid exploration of the chemical spaces targeting the desired 
+functionalities. Here we introduce a novel approach for the active learning over the chemical 
+spaces based on hypothesis learning. We construct the hypotheses on the possible relationships 
+between structures and functionalities of interest based on a small subset of data and introduce 
+them as (probabilistic) mean functions for the Gaussian process. This approach combines the 
+elements from the symbolic regression methods such as SISSO and active learning into a single 
+framework. Here, we demonstrate it for the QM9 dataset, but it can be applied more broadly to 
+datasets from both domains of molecular and solid-state materials sciences. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+email: ghosha@ornl.gov 
+ 
+
+Introduction 
+Chemical discovery1-4 is rooted in quantitative structure-activity/property relationships 
+(QSAR/QSPR).5-12 These efforts primarily rely on finding appropriate representation of molecules 
+followed by establishing relationships between structure and activity they exhibit. These models 
+are harnessed to explore chemical space to select molecules of interest 13-15 for drug targets,16-20 
+antibiotics,21 catalysts,22-23 photovoltaics,24-27 organic electronics,28 redox-flow batteries.29 In 
+addition, chemical discovery also includes understanding of chemical processes such as reaction 
+energy pathways,30-32 optimization of reaction conditions,33 (for e.g., catalytic activity34), 
+crystallization,35-36 docking,37 and synthesis.38-39  
+The QSAR/QSPR techniques have proven to be useful in all (not limited to) such scenarios. 
+The popularity40-42 remains in their simplicity to incorporate structural information combined with 
+physicochemical properties, reliability to capture the property landscape, capability to identify 
+existing chemical patterns, and identify activity cliffs within the data while being computationally 
+affordable to perform. The descriptors or features can be multi-dimensional descriptors capturing 
+electronic or topological characteristics. Alternatively, these can be fingerprints that are the 
+effective representations of molecules via graph-based or string representations (SMILES,43 
+SELFIES44). QSAR/QSPR models began its journey almost 60 years ago with the seminal work 
+lead by Hansch et al.45 in which a few simple descriptors were used to capture a 2D structure-
+activity relationship. Since then, this field has seen a steep rise in utilization of variety of traditional 
+ML algorithms (Naïve Bayes, Support Vector Machine, Random Forest, to name a few) for 
+property/process prediction followed by validation by experimental synthesis. Its success is also 
+credited to generation of easily accessible public repositories (e.g., PubChem,46-48 ZINC,49 
+ChEMBL,50-51 QM9,52 ANI-1x,53 and QM7-X54) containing structural and physiochemical 
+properties (computed with quantum mechanical calculations or observed with experiments) on 
+thousands to millions of molecules. Altogether these have paved the path forward in chemical 
+design, discovery with day-to-day applications. 
+If QSAR/QSPR studies have created a revolution in in silico design efforts, applications of 
+deep neural network (NN) algorithms55-56 have further accelerated this progress. They have 
+enabled efficient usage of the big data for not only finding molecules of interest but also quantify57-
+60 molecular interactions, chemical bonding, inverse design of molecules for targets and gain novel 
+insights into mechanisms. However, the performance of any of these models is highly dependent 
+on the quality, quantity, modelability of the datasets. Furthermore, a discovery process necessitates 
+extrapolation of learned correlative relationships onto the previously unseen regions of chemical 
+space.  
+Correspondingly, the task of generation61 of new molecules by going beyond the standard 
+(manual) design rules or solutions has gained much attention. Several studies62 have been reported 
+where different NN-based algorithms are explored to accomplish this challenge. Autoencoders are 
+one of the common models that are being used to encode molecules63 via complex latent 
+representations to optimize for specific properties and map them back to molecular structures 
+through decoding. Another set of examples include applications of recurrent NN64 algorithms 
+where molecule generation is treated as a sequencing task and the algorithm is permitted to 
+
+generate samples at each stage, as informed by the model. There are also studies on using self-
+attention driven transformer models65-66 for targeted structure generation. In addition, 
+reinforcement learning strategies67-68 have been implemented in this context where molecules with 
+multiple target objectives can be found. The modern molecular generative models have 
+transformed standard string representations of molecules towards embedded spaces69 with 
+information on the entire molecular scaffold.  
+However, the learning approach behind most of the generative models traverse through 
+latent embeddings. The latter are generally not smooth, precluding direct gradient-based 
+optimization methods. Once trained over full libraries of molecules, the fraction of the space 
+occupied with molecules with useful functionalities is typically small, making their discovery 
+complex. The chemical space is non-differentiable, precluding the gradient-based descent or 
+simple Gaussian processes (GPs)70-71 based methodologies. ML methods including variational 
+autoencoders aim to construct a suitable low-dimensional and ideally differentiable latent 
+embeddings for the chemical space, allowing for the Bayesian optimization (BO)72-74 type 
+processes. However, these methodologies to date have been based on either construction of the 
+embedding space for the full library of candidate molecules or finding similar kernel of 
+representation for these molecules for target explorations.  
+Historically, Gaussian process (GPs) has been used within the active learning and BO, 
+making these processes purely data-driven and non-parametric in nature. It interpolates functional 
+behavior over relatively low-dimensional parameters space. In a classical GP, a kernel function 
+(such as radial basis function kernel) is utilized to define the degree of correlation across the 
+parameters space. The kernel parameters are inferred based on the available data, obtained during 
+exploration-exploitation steps. It does not incorporate any prior information of physical or 
+chemical behavior of the system in the process. It learns the physics of the system from the data 
+itself via kernel function with possibilities of leading to suboptimal results. Consequently, the 
+number of optimization steps necessary to reconstruct functional behavior, even scanning over low 
+parameters space becomes large. More advanced physics-informed kernel functions75-77 may help 
+in such cases which is an area of active research. However, proper application of this technique in 
+the domain of chemical or physical sciences demands going above the usage of conventional GP 
+in BO framework to model functionalities of interest.78-81 A series of molecular kernels82 such as 
+fingerprint kernels (e.g., scalar product kernel, Tanimoto kernel), string kernels (e.g., kernels based 
+on SMILEs, SELFIEs) and graph kernels (typically uses molecular fragments) can be used in GP 
+framework. Once the best-performing kernel is chosen, BO is performed for applications in real 
+world scenarios such as optimization of chemical reactions.  
+In a recent work, Ziatdinov et al.83-85 introduced a physics-augmented algorithm for active 
+learning and Bayesian optimization. It combines the flexibility of GP models with physical priors 
+allowing for the hypothesis-driven discovery in ML. To date, it has been applied86 to the 
+experiments in scanning probe microscopy, providing new insights into the concentration-induced 
+phase transition and identifying domain growth laws in ferroelectric materials. 
+Here, we extend the concept of hypothesis learning to molecular discovery. We combine 
+it with the compressed-sensing methodologies for identifying relevant structural descriptors and 
+
+evaluate multiple automatically generated hypotheses with a reward-driven acquisition (similar to 
+the reinforcement learning) to select next evaluation points. Specifically, the hypotheses are 
+selected using the compressed sensing performed on combinations of nonlinear functionalized 
+features to find a list of the most relevant combinations. This step is followed by balancing 
+dimensions (i.e., respecting physics constraints) with respect to the target property to formulate 
+them into feasible equations. Here, we only consider a handful of easily computable features 
+related to property of interest, to keep the hypotheses in a simple form that is easy to calculate. 
+Finally, we evaluate the hypotheses over a wide parameters space to predict functionalities of 
+interest within the active learning loop. We have utilized the QM9 dataset on isolated molecules 
+as a use case to establish this tail of our work.  
+A schematic of the generalized framework detailing the active cruise between design and 
+discovery using hypothesis learning is shown in Figure 1. The results along with estimated 
+uncertainties show a generalized cost-effective way to approximate structure-property relations, 
+applicable to a wide variety of material systems.  
+ 
+Figure 1: Schematic of workflow, from design to discovery. Figure (left panel) shows 
+commonly used simulation techniques to generate reliable data for various materials systems. The 
+right panel establishes the active learning loop - combining features to come up with mathematical 
+formulations as statistically derived scientific hypotheses, to be evaluated for discovery structure-
+property relations. 
+ 
+Coarse-grained MD
+Ab-initio MD
+Atomistic MD
+Quantum 
+Mathematical 
+formulations
+Selection & 
+combination of 
+physical 
+descriptors
+Features 
+Experiment
+Physics-informed 
+featurization & sparsification
+Initialization
+Scientific 
+hypothesis
+!"#$!⋯# → &!⋯#
+Exploration
+
+Results and Discussion 
+General Considerations: 
+The physics-informed featurization scheme that we designed is built upon the compressed 
+sensing methodologies utilized by Ghiringhelli et al.87 for features selection, implemented here as 
+the seed step for the discovery cycle of an active learning process. Similar to the original SISSO 
+implementation, our physics-informed featurization and sparsification scheme allows for the 
+selection of the most relevant descriptors which is obtained by using the least absolute shrinkage 
+and selection operator (LASSO). The LASSO algorithm employed as a part of feature selection 
+scheme uses the sparsity of the l1 norm to effectively reduce a descriptor set to the most relevant 
+descriptors (di) contained in full set (D). It selects the non-zero terms of the l1 regularized linear 
+least squares approximation of the target property (P). The target property is approximated as P(d) 
+= dc, where c is the coefficient (or weight) associated with Ω dimensional descriptor d. The 
+solution to this equation can be determined by minimizing the argmin (||𝑃 − 𝐷𝑐||!
+!) + 𝜆||𝑐||". 
+The coefficient c is non-zero for all featurized descriptor which is then ranked to determine the 
+corresponding importance. 
+The generation of nonlinear combinations of descriptors (di) by applying several mathematical 
+operators such as, 1/x, √x, x2, x3, log(x), 1/ log(x), and exp(x), on each feature allows to form a 
+nonlinear mapping between D and P. In addition, the complexity by combining these 
+functionalized descriptors via summation allows to generate a more effective map between D and 
+P. We have considered functionalized descriptors utilizing 2 or 3 terms for the purpose of this 
+study. Here, we note that inclusion of more terms may lead to more accurate correlation to 
+endpoint. However, it also introduces additional uncertainty carried by each of the terms combined 
+with mathematical operators. The physics-informed featurization and sparsification method allows 
+us to combine multiple features in a linear combination to establish direct correlation to the 
+endpoint target. Within this method, we search over a large combinatorial space, combine features 
+followed by balancing units/dimensionality (with coefficients) to convert them into feasible 
+equations. These are the mathematical formulations that are then turned into probabilistic models 
+(hypotheses) by introducing suitable priors on parameters, applicable to all use cases.  
+ 
+The second element of the proposed approach is the hypothesis-driven active learning built 
+upon SISSO-derived functional forms. In the hypothesis learning, we utilize a structured GP (sGP) 
+as opposed to standard zero mean GP as our surrogate model(s) to insert physics-informed priors 
+in the GP/BO framework. To illustrate this approach, we note that in the conventional GP/BO 
+process, GP is defined as 
+ 
+ 
+ 
+ 
+𝑦 = 𝑓(𝑥) + e, 
+𝑓 ~ 𝑀𝑉𝑁𝑜𝑟𝑚𝑎𝑙	(𝑚, 𝐾)  
+( 1) 
+where MVNormal is a multivariate normal distribution, m is a prior mean function typically set to 
+0, 𝐾 is a prior covariance functions (kernel), and e is a normally distributed observational noise.  
+ 
+The training process of GP model involves inferring kernel parameters given the available 
+set of observations (x, y) using Bayesian inference techniques. Once the training is completed, the 
+
+probabilistic predictions of the function over the unmeasured parameter space can be obtained by 
+sampling from a distribution: 
+ 
+𝑓∗	~	𝑀𝑉𝑁𝑜𝑟𝑚𝑎𝑙8𝜇$
+%&'(, Σ$
+%&'(< 
+( 2) 
+𝜇$
+%&'( = 𝑚(𝑋∗) + 𝐾(𝑋∗, 𝑋|𝜃)𝐾(𝑋, 𝑋|𝜃))"8𝑦 − 𝑚(𝑋)<, 
+Σ$
+%&'( = 𝐾(𝑋∗, 𝑋∗|𝜃) − 𝐾(𝑋∗, 𝑋|𝜃)(𝑋, 𝑋|𝜃))"𝐾(𝑋, 𝑋∗|𝜃) 
+( 3) 
+Here new inputs are denoted by 𝑋∗. We can obtain a posterior predictive distribution for each set 
+of kernel parameters. The next point to evaluate is then determined by 
+𝑥*+,( = arg 𝑚𝑎𝑥,
+1
+𝑀 D 𝛼F𝜇$!
+%&'(, Σ$!
+%&'(G
+-
+./"
+ 
+( 4) 
+Here 𝛼 is a pre-defined acquisition function and M is the number of posterior samples with kernel 
+parameters. 
+ 
+Within the sGP, the prior mean function in Equation 1 is substituted by a physics-informed 
+probabilistic model whose parameters are inferred jointly with the kernel parameters. The posterior 
+mean function in Equation 3 then becomes 
+𝜇$!0!
+%&'( = 𝑚8𝑋∗|𝜙.< + 𝐾8𝑋∗, 𝑋|𝜃.<𝐾8𝑋, 𝑋|𝜃.<
+)" F𝑦 − 𝑚8𝑋|𝜙.<G 
+( 5) 
+ 
+Hence, the active learning scheme with sGP incorporates the uncertainty associated with 
+the model parameters. The parametric nature of the model captures the physics-based knowledge 
+to determine the next point of exploration.  
+ 
+The hypothesis-driven learning utilizes an ensemble of the sGP models that are effectively 
+competing to reconstruct physical behavior over a large parameters space. The prior mean 
+functions are derived from SISSO method. We utilize Matern kernel for all the sGP models and 
+use the predictive uncertainty as the acquisition function, 𝛼 = diag(Σ$
+%&'() A basic reinforcement 
+learning policy (epsilon-greedy) is used to sample a hypothesis at each exploration step. The 
+sampled hypothesis gets wrapped into sGP whose parameters are inferred using Hamiltonian 
+Monte Carlo. The posterior distributions over the sGP parameters are used to obtain a probabilistic 
+prediction over the unmeasured parts of the parameter space. The total predictive uncertainty is 
+then compared with that from a previous step, and the sampled model gets rewarded (penalized) 
+if the uncertainty decreased (increased).  
+ 
+ 
+
+Formation enthalpy of molecules: 
+Predicting molecular energetics to understand formability and stability of isolated 
+molecules have become one of the top applications of modern day’s ML models. These molecules 
+play key role in both materials design and drug discovery. Quantum mechanical (QM) calculations 
+within various approximations serve as the basis for studying underlying chemistry, finding active 
+sites to understanding chemical reactions. Among various freely accessible online datasets owning 
+such information, QM9 is popular among researchers due to data homogeneity, purity, and lack of 
+noise. As a result, it has turned out to be a classical benchmark for testing various ML models. 
+QM9 dataset contains information on geometric, energetic, electronic, and thermodynamic 
+properties of ~130,000 organic compounds with elements C, H, O, N and F, as computed with 
+molecular DFT with B3LYP/6- 31G (2df, p) level of theory. Common ML algorithms such as 
+linear regression, kernel ridge regression (KRR), random forest regression and deep neural 
+networks88 have been used to predict atomization energy,89 HOMO-LUMO energy,90 formation 
+energy.91 
+ 
+ 
+ 
+ 
+Figure 2: Flow diagram for QM9 dataset. Figure shows the key stages of the framework as 
+implemented for the QM9 dataset. The key steps include generation of a partial dataset with a 
+combination of a few features from the original QM9 dataset, followed by formulation of scientific 
+hypotheses by physics-informed featurization method which get evaluated in the reward-driven 
+hypothesis-driven learning scheme. 
+ 
+Proper benchmarking investigations,92-99 comparing computational accuracy, time, convergence 
+of algorithms, starting from traditional ML to state-of-the-art neural networks (NNs) are also 
+available in literature. Most of these studies are dependent on either high dimensional feature space 
+or molecular representations to reach target property. In general, an 80/20 split between the training 
+and test is assumed to perform the benchmarking tasks while drawing conclusions on existing 
+structure-property relations. 
+ 
+Despite the extensive effort, it has not been clearly understood how to establish structure-
+property relationships on unknown chemical space (even within QM9). In other words, for learning 
+about new molecules and their corresponding properties - which is where capability of ML tools 
+can really be tested - we also need to learn physical or phenomenological laws relating variables 
+with the target of interest on-the-fly in the data-efficient manner. 
+ 
+QM9 dataset
+Partial QM9 dataset 
+with few features
+Generate 
+Hypotheses
+Active learning for 
+entire dataset
+
+ 
+Figure 3: Distribution and prediction of formation enthalpy (Hartree/g/mol). (a) Comparative 
+plot and (b) histograms of probability density of formation enthalpy for the entire QM9 dataset 
+superimposed with that of 1000 molecules, used in hypotheses generation. (b) Average rewards 
+for each model predicting formation enthalpy in active learning scheme. 
+ 
+In this example, we lay out how we can employ hypothesis-driven learning to reconstruct 
+behavior of formation enthalpy (FE, Hartree/g/mol) at 298.15 K with respect to topological or 
+(c)
+(a)
+(b)
+
+4.0
+QM9 dataset
+Molecules for hypotheses generation
+3.5
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+-2.5
+-5.0
+-4.5
+-4.0
+-3.5
+-3.0
+Enthalpy (Hartree/g/mol)QM9 full dataset
+Molecules for hypotheses generationpolar surface area (TPSA, Å2) for QM9 dataset. A schematic of the framework as implemented for 
+this example is shown in Figure 2. We select the first 1,000 molecules from the QM9 dataset and 
+create a feature-target dataset with a handful of features such as molecular weight (MW), polar 
+surface area (TPSA), molar log P (molelogP), spatial extent (SP) and internal energy (IE). Features 
+such as MW, TPSA, molelogP are computed (easy and computationally cheap to obtain) using the 
+python implementation of RdKit package. The chemical properties such as SP and IE are directly 
+available from QM9 dataset, which are otherwise are calculated using computationally expensive 
+DFT methods. 
+The general chemistry (as described by the features computed with structural information) 
+of the first 1000 molecules in QM9 largely differ from what is encompassed by the entire dataset. 
+Therefore, the challenge is to find if simple mathematical expressions generated with a small 
+percentage of QM9 are capable to be extended for the remaining data set to yield meaningful 
+predictions of FE. A plot showing distribution of FE of the entire dataset containing 133885 
+molecules as compared to 0.74% of the dataset as used for generating hypotheses are shown in 
+Figure 3 (a). The normalized histogram plots of FE in Figure 3 (b) show how the first set of 1000 
+molecules represent the full dataset. We perform the physics-informed featurization to draw three 
+different competing hypotheses as mentioned below.  
+ 
+𝐹𝐸 = 𝐼𝐸 ∗ O1 + P𝑇𝑃𝑆𝐴
+𝑆𝑃 T
+!
+U 
+( 6) 
+𝐹𝐸 = 𝐼𝐸 ∗ O1 + P𝑇𝑃𝑆𝐴
+𝑆𝑃 T
+!
++ 𝑚𝑜𝑙𝑒𝑙𝑜𝑔𝑃!U 
+( 7) 
+𝐹𝐸 = 𝐼𝐸 ∗ P1 + 𝑚𝑜𝑙𝑒𝑙𝑜𝑔𝑃
+1 + 𝑇𝑃𝑆𝐴T 
+( 8) 
+ 
+We have considered priors for all parameters as listed below in Table 1. Next, we 
+implement the hypothesis-driven learning scheme by treating the Equations 6-8 as individual 
+probabilistic models wrapped into sGP. 
+Table 1: Table lists priors considered for all parameters listed in the equations for predicting 
+excitation energy. 
+Parameter 
+Prior 
+IE (Hartree/g/mol) 
+Uniform [-4, 2] 
+SP (Å2) 
+Uniform [2, 0.05] 
+ 
+Molecules utilized at the stage of hypotheses generation are discarded from training and testing. 
+We use 300 samples picked randomly from the dataset as seeds or starting points. Five different 
+set of samples are initialized in this manner. The average reward over all initializations for each 
+
+model after 200 explorations is shown in Figure 3 (c). Model 2 appears to have accumulated the 
+highest reward to reconstruct FE for the entire chemical space of QM9.  
+However, if we look at individual runs as plotted in Figure 4, it is evident that Model 1 and 
+Model 2 compete during exploration. The median uncertainty (a, c, e) and reward (b, d, f) for each  
+ 
+Figure 4: Hypothesis-driven active learning with structured Gaussian processes (sGPs) for 
+entire QM9 dataset. The median uncertainty (a, c, e) and reward (b, d, f) for each model with 
+respect to explorations step for three different random initializations are shown here. 
+ 
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+
+model 1
+model 2
+model 3model 1
+model 2
+model 3model 1
+model 2
+model 3model with respect to exploration steps for three different random initializations are plotted in 
+Figure 4. The additional jumps in the uncertainty during a segment of exploration steps can be 
+attributed to structural complexity, evolution of the power laws during the learning process. In 
+addition, the approximations given by Model 1 and Model 2 have similar functional forms, due to 
+which, both lead to reasonable predictions. Model 3 always collects the least amount of reward 
+indicating failure of this specific approximation to capture the distribution of FE over the full 
+dataset. The approximations given by the models may slightly vary depending on the initial set of 
+molecules on which the hypotheses are generated. These variations can still be captured by going 
+to higher order approximations during formulation of model expressions which is expected to 
+include combination of the already utilized terms in our present models. 
+This example establishes the capability of this technique to reconstruct behavior of a 
+molecular property and how it varies over a wide variety of chemical space. It only utilizes a 
+handful of easily computable features along with simplistic (more interpretable) mathematical 
+formulations, leading to meaningful predictions. 
+In summary, we have presented an example for molecules in QM9 dataset to showcase the 
+capability of physics-informed featurization combined with hypothesis-driven active learning for 
+reconstruction of materials property. It helps to understand and approximate the functional 
+behavior of systems belonging to different materials class for which data from simulations or 
+experiments may not be available. The framework proposed in this work allows for a couple of 
+prominent advances in the physical science and ML community. First is its potential to serve as a 
+template to come up with easily interpretable models to represent functionalities, obtained from 
+previous observations. It also attempts to meet both design and discovery requirements, truly 
+needed for facilitating progresses in physical sciences. 
+ 
+Acknowledgements 
+This research is sponsored by the INTERSECT Initiative as part of the Laboratory Directed 
+Research and Development Program of Oak Ridge National Laboratory, managed by UT-Battelle, 
+LLC, for the U.S. Department of Energy under contract DE-AC05-00OR22725. Part of this 
+research was conducted at the Center for Nanophase Materials Sciences, which is a DOE Office 
+of Science User Facility. 
+ 
+Author contributions 
+A.G. proposed, designed the workflow, and wrote the draft of the manuscript. S.V.K. discussed 
+the concept and participated in writing. M.Z. developed the structured GP approach and oversaw 
+the preparation of the manuscript. 
+Data Availability 
+All datasets as utilized in this work can be accessed via the github repository. 
+https://github.com/aghosh92/SISSO_sGP 
+
+ 
+Code Availability 
+The workflow can be accessed via the github repository. https://github.com/aghosh92/SISSO_sGP 
+ 
+Conflict of interest 
+The authors declare no competing interests. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
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+

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@@ -0,0 +1,1398 @@
+arXiv:2301.00341v1  [math.CO]  1 Jan 2023
+Symmetric polynomials connecting unsigned
+and signed relative derangements
+Ricky Xiao-Feng Chen, Yu-Chen Ruan
+School of Mathematics, Hefei University of Technology
+Hefei, Anhui 230601, P. R. China
+xiaofengchen@hfut.edu.cn, 1059568476@qq.com
+Abstract
+In this paper, we introduce polynomials (in t) of signed relative derangements
+that track the number of signed elements. The polynomials are clearly seen to be in
+a sence symmetric. Note that relative derangements are those without any signed
+elements, i.e., the evaluations of the polynomials at t = 0. Also, the numbers of
+all signed relative derangements are given by the evaluations at t = 1. Then the
+coefficients of the polynomials connect unsigned and signed relative derangements
+and show how putting elements with signs affects the formation of derangements.
+We first prove a recursion satisfied by these polynomials which results in a recursion
+satisfied by the coefficients. A combinatorial proof of the latter is provided next. We
+also show that the sequences of the coefficients are unimodal. Moreover, other results
+are obtained, for instance, a kind of dual of a relation between signed derangements
+and signed relative derangements previously proved by Chen and Zhang is presented.
+Keywords: Derangements, Relative derangements, Symmetric polynomials, Uni-
+modal
+Mathematics Subject Classifications: 05C05, 05A19, 05A15
+1
+Introduction
+A derangement on a set [n] = {1, 2, . . ., n} is a permutation π = π1π2 · · · πn on [n] such
+that πi ̸= i for all i ∈ [n], i.e., a permutation without fixed points. We use Dn to denote
+the set of derangements on [n] and Dn to denote the number of derangements on [n].
+The study of derangements may date back to Euler who showed that the probability for
+a random permutation to be a derangement tends to 1/e. It is also well known (e.g.,
+Stanley [8, Chapter 2]) that
+Dn = (n − 1)(Dn−1 + Dn−2).
+(1)
+1
+
+A relative derangement π = π1π2 · · ·πn on [n] is a permutation such that πi+1 ̸= πi + 1
+for 1 ≤ i ≤ n − 1. Let Qn denote the set of relative derangements on [n] and Qn = |Qn|.
+With the aid of the notion of skew derangements, Chen [4] combinatorially showed that
+Qn = Dn + Dn−1.
+(2)
+A signed permutation π on [n] can be viewed as a bijection on the set [n] �{1, . . . , n}
+such that π(i) = π(i), where j = j.
+Intuitively, a signed permutation on [n] is just
+an ordinary permutation π = π1π2 · · · πn with some elements associated with a bar. For
+example, 1342 is a signed permutation on {1, 2, 3, 4}. These elements with a bar are called
+signed elements or bar-elements. The set of signed permutation on [n] is often denoted
+by Bn. A signed derangement [1] on [n] is a signed permutation π = π1π2 · · · πn such that
+πi ̸= i, for all i ∈ [n]. For example, 1342 is a signed derangement in B4, whereas 1342
+is not since it has a fixed point 1. A signed relative derangement (or sometimes called
+relative derangement of type B, see [5]) on [n] is a signed permutation on [n] such that
+i is not followed by i + 1, and i is not followed by i + 1. For example, 1324 is a signed
+relative derangement. We denote by DB
+n and QB
+n the sets of signed derangements and
+signed relative derangements on [n], respectively. Let DB
+n = |DB
+n | and QB
+n = |QB
+n |. Based
+on the notion of signed skew derangements, Chen and Zhang [5] proved that
+QB
+n = DB
+n + DB
+n−1.
+(3)
+One of our results in this paper is a kind of dual of this relation, that is, we present a
+relation expressing DB
+n in terms of fn that counts an essential subset of sequences in QB
+n .
+Obviously, the subset of sequences with zero signed elements is Qn and hence Qn ⊂ QB
+n .
+It is natural to consider the subset consisting of sequences with m signed elements. As
+such, a polynomial tracking the number of signed elements is introduced. While many
+polynomials or q-analogues associated to derangements have been studied, for instance,
+the q-enumeration of derangements in Bn by flag major index [1], the excedances of
+derangements [6,10], the q-enumeration of derangements by major index [9], and the cyclic
+polynomials of derangements [7], our polynomials here seem to have been overlooked. In
+addition, our polynomials have a nice property, namely, they are in a sense symmetric.
+The paper is organized as follows. In Section 2, we introduce the symmetric poly-
+nomials and prove a recursion satisfied by them. Various results are then derived as a
+consequence. Section 3 is devoted to presenting a combinatorial proof of the resulting
+recursion satisfied by the coefficients as well as proving a unimodality property.
+2
+Symmetric polynomials
+Let b(π) be the number of signed elements in π ∈ QB
+n . The polynomial of signed relative
+derangements recording the number of signed elements is then given by
+QB
+n (t) =
+�
+π∈QB
+n
+tb(π) =
+n
+�
+m=0
+qn,mtm,
+2
+
+where qn,m denotes the number of signed relative derangements with exactly m signed
+elements.
+It is evident that qn,m = qn,n−m as we can obtain a signed relative derangment with
+n − m bar-elements by turning a signed element into its unsigned counterpart and vice
+versa. Therefore, the polynomial QB
+n (t) is in a sense symmetric.
+Denote by �QB
+n the set of signed permutations on the set [n] where in each signed
+permutation two consecutive entries of the form i(i+ 1) or i(i + 1) for some 1 ≤ i < n−1
+appears exactly once. For example, 4231 ∈ �QB
+4 .
+For π ∈ QB
+n , we denote the resulting sequence from removing n or n whichever appears
+in π by π↓. The following lemma should not be hard to observe.
+Lemma 1. For any π ∈ QB
+n , we have either π↓ ∈ QB
+n−1 or π↓ ∈ �QB
+n−1.
+Accordingly, we immediately have
+QB
+n (t) =
+�
+π∈QB
+n
+tb(π) =
+�
+π∈QB
+n , π↓∈�QB
+n−1
+tb(π) +
+�
+π∈QB
+n , π↓∈QB
+n−1
+tb(π).
+(4)
+To obtain a recursion of QB
+n (t), we next study the two sums on the right-hand side of
+eq. (4) in detail. For π = π1π2 · · · πn−1 ∈ QB
+n−1 and n ≥ 2, denote by S↑(π) the set
+of sequences in �QB
+n that result from π by lifting the elements larger than πi (for some
+1 ≤ i ≤ n − 1) by one and replacing πi with a length-two sequence πi(πi + 1), where we
+define the addition for bar-elements by the rule i + 1 = i + 1. Moreover, if an element x
+appears an entry in π, we write x ∈ π.
+Lemma 2. For n ≥ 3 and any π ∈ QB
+n , we have
+�
+π′∈S↑(π)
+tb(π′) = b(π)tb(π)+1 + (n − b(π))tb(π).
+(5)
+Proof. For any π = π1π2 · · ·πn ∈ QB
+n , it has b(π) bar-elements and n − b(π) elements
+without a bar. For any πi ∈ π with a bar, it will generate an additional bar-element after
+lifting the elements larger than πi (for some 1 ≤ i ≤ n − 1) by one and replacing πi with
+a length-two sequence πi(πi + 1). In other words, it will contribute tb(π)+1. However, for
+any πi ∈ π without a bar, the number of bar-elements in the sequence will not change.
+Therefore, it contributes tb(π). Summarizing the two cases gives the lemma.
+The lemma right below is not difficult to verify.
+Lemma 3. If π, π′ ∈ QB
+n−1 and π ̸= π′, then S↑(π) � S↑(π′) = ∅. Moreover,
+�QB
+n =
+�
+π∈QB
+n−1
+S↑(π).
+(6)
+3
+
+Proposition 4. For n ≥ 3, we have
+�
+π∈QB
+n , π↓∈�QB
+n−1
+tb(π) = (1 + t)
+�
+(t2 − t)QB
+n−2
+′(t) + (n − 2)QB
+n−2(t)
+�
+,
+(7)
+where QB
+n
+′(t) stands for the derivative of QB
+n (t) with respect to t.
+Proof. First, by construction, there are exactly two signed permutations π, π′ ∈ QB
+n such
+that π↓ = π′↓ ∈ �QB
+n−1, and vice versa, where if n ∈ π then π′ can be obtained by replacing
+n with n in π. Thus, tb(π↓) = tb(π′↓) = tb(π) = tb(π′)−1 and
+�
+π∈QB
+n , π↓∈�QB
+n−1
+tb(π) =
+�
+π′∈�QB
+n−1
+(1 + t)tb(π′).
+Next, we have
+�
+π′∈�QB
+n−1
+tb(π′) =
+�
+π′′∈QB
+n−2
+�
+π′∈S↑(π′′)
+tb(π′)
+=
+�
+π′′∈QB
+n−2
+�
+b(π′′) · t +
+�
+n − 2 − b(π′′)
+��
+tb(π′′)
+=
+�
+π′′∈QB
+n−2
+�
+(t − 1)b(π′′)tb(π′′) + (n − 2)tb(π′′)�
+= (t2 − t)QB
+n−2
+′(t) + (n − 2)QB
+n−2(t),
+where the first two equalities follow from Lemma 3 and Lemma 2, respectively, and then
+the proof follows.
+Proposition 5. For n ≥ 3, we have
+�
+π∈QB
+n , π↓∈QB
+n−1
+tb(π) = (nt + n − 1)QB
+n−1(t) + (1 − t)
+�
+π′∈QB
+n−1, n−1∈π′
+tb(π′).
+(8)
+Proof. A sequence π ∈ QB
+n where n appears can be clearly obtained by inserting n into a
+sequence π↓ ∈ QB
+n−1. We distinguish two cases:
+• if n − 1 appears in π↓ ∈ QB
+n−1, there are n − 1 positions where n can be inserted.
+• if n − 1 appears in π↓ ∈ QB
+n−1, there are n positions where n can be inserted.
+Note that in both cases, we have b(π) = b(π↓). Thus,
+�
+π∈QB
+n , π↓∈QB
+n−1, n∈π
+tb(π) =
+�
+π′∈QB
+n−1, n−1∈π′
+(n − 1) · tb(π′) +
+�
+π′∈QB
+n−1, n−1∈π′
+n · tb(π′)
+= (n − 1)QB
+n−1(t) +
+�
+π′∈QB
+n−1, n−1∈π′
+tb(π′) .
+Similarly, the situation of inserting n can be calculated. We also distinguish two cases:
+4
+
+• if n − 1 appears in π↓ ∈ QB
+n−1, there are n positions where n can be inserted.
+• if n − 1 appears in π↓ ∈ QB
+n−1, there are n − 1 positions where n can be inserted.
+The difference is that in this case, we have b(π) = b(π↓) + 1. Thus,
+�
+π∈QB
+n , π↓∈QB
+n−1, n∈π
+tb(π) =
+�
+π′∈QB
+n−1,n−1∈π′
+nt · tb(π′) +
+�
+π′∈QB
+n−1,n−1∈π′
+(n − 1)t · tb(π′)
+= ntQB
+n−1(t) − t
+�
+π′∈QB
+n−1,n−1∈π′
+tb(π′).
+Combining the above two cases, we obtain the proposition.
+Proposition 6. For n ≥ 3, we have
+�
+π′∈QB
+n−1, n−1∈π′
+tb(π′) =(n − 1)tQB
+n−2(t) + t
+�
+(t2 − t)QB
+n−3
+′(t) + (n − 3)QB
+n−3(t)
+�
+− t
+�
+π′′∈QB
+n−2, n−2∈π′′
+tb(π′′) .
+(9)
+Proof. Analogously, we first have
+�
+π′∈QB
+n−1, n−1∈π′
+tb(π′) =
+�
+π′∈QB
+n−1, n−1∈π′, π′↓∈QB
+n−2
+tb(π′↓) +
+�
+π′∈QB
+n−1, n−1∈π′, π′↓∈�QB
+n−2
+tb(π′↓).
+The first sum of the right-hand side has been obtained in Proposition 5 and equals
+(n − 1)tQB
+n−2(t) − t
+�
+π′′∈QB
+n−2, n−2∈π′′
+tb(π′′).
+Following the proof of Proposition 4, the second sum of the right-hand side equals
+�
+π′∈�QB
+n−2
+t · tb(π′) = t
+�
+π′′∈QB
+n−3
+�
+π′∈S↑(π′′)
+tb(π′)
+= t
+�
+π′′∈QB
+n−3
+�
+b(π′′) · t +
+�
+n − 3 − b(π′′)
+��
+tb(π′′)
+= t
+�
+π′′∈QB
+n−3
+�
+(t − 1)b(π′′)tb(π′′) + (n − 3)tb(π′′)�
+= t
+�
+(t2 − t)QB
+n−3
+′(t) + (n − 3)QB
+n−3(t)
+�
+.
+The rest is clear and the proof follows.
+Based on Proposition 4–6, we conclude
+5
+
+Theorem 7. For n ≥ 3, the following holds
+QB
+n (t) = (n − 1)(t + 1)QB
+n−1(t) +
+�
+(3n − 5)t + (n − 2)
+�
+QB
+n−2(t)
++ (t3 − t)QB
+n−2
+′(t) + (2n − 6)tQB
+n−3(t) + 2t2(t − 1)QB
+n−3
+′(t),
+(10)
+and QB
+0 (t) = 0, QB
+1 (t) = 1 + t, QB
+2 (t) = t2 + 4t + 1.
+Proof. According to Proposition 4–6, we first obtain
+QB
+n (t) =
+�
+π∈QB
+n
+tb(π) =
+�
+π∈QB
+n , π↓∈�QB
+n−1
+tb(π) +
+�
+π∈QB
+n , π↓∈QB
+n−1
+tb(π)
+=(1 + t)
+�
+(t2 − t)QB
+n−2
+′(t) + (n − 2)QB
+n−2(t)
+�
++ (nt + n − 1)QB
+n−1(t) + (1 − t)
+�
+π′∈QB
+n−1
+n−1∈π′
+tb(π′)
+=(1 + t)
+�
+(t2 − t)QB
+n−2
+′(t) + (n − 2)QB
+n−2(t)
+�
++ (nt + n − 1)QB
+n−1(t)
++ (1 − t)
+�
+(n − 1)tQB
+n−2(t) + t
+�
+(t2 − t)QB
+n−3
+′(t) + (n − 3)QB
+n−3(t)
+�
+− t
+�
+π′′∈QB
+n−2
+n−2∈π′↓
+tb(π′↓) �
+.
+Iterating using Proposition 6 and using the fact that
+�
+π′∈QB
+1 , 1∈π′
+tb(π′) = t, we have
+QB
+n (t) =(nt + n − 1)QB
+n−1(t) + (1 − t)
+� n−2
+�
+k=1
+(−1)k+1(n − k)tkQB
+n−k−1(t)
+�
++ (−1)n(1 − t)tn−1 + (1 + t)
+�
+(t2 − t)QB
+n−2
+′(t) + (n − 2)QB
+n−2(t)
+�
++ (1 − t)
+�n−2
+�
+k=1
+(−1)k+1tk�
+(t2 − t)QB
+n−k−2
+′(t) + (n − k − 2)QB
+n−k−2(t)
+�
+�
+=(nt + n − 1)QB
+n−1(t) + (2n − 4)QB
+n−2(t) + (−1)n(1 − t)tn−1
++
+n−2
+�
+k=1
+(−1)ktk−1�
+(n − k − 1) + (2k + 1 − 2n)t + (n − k)t2�
+QB
+n−k−1(t)
++ (t3 − t)QB
+n−2
+′(t) +
+n−2
+�
+k=1
+(−1)k+1tk+1(2t − 1 − t2)QB
+n−k−2
+′(t).
+(11)
+Consequently, we have
+QB
+n−1(t) =
+�
+(n − 1)t + n − 2
+�
+QB
+n−2(t) + (2n − 6)QB
+n−3(t) + (−1)n−1(1 − t)tn−2
++
+n−3
+�
+k=1
+(−1)ktk−1�
+(n − k − 2) + (2k + 3 − 2n)t + (n − k − 1)t2�
+QB
+n−k−2(t)
++ (t3 − t)QB
+n−3
+′(t) +
+n−3
+�
+k=1
+(−1)k+1tk+1(2t − 1 − t2)QB
+n−k−3
+′(t).
+6
+
+Then, it is observed that the two sums in the last expression of eq. (11) equals
+(−t)
+�
+QB
+n−1(t) −
+�
+(n − 1)t + n − 2
+�
+QB
+n−2(t)
+− (2n − 6)QB
+n−3(t) − (−1)n−1(1 − t)tn−2 − (t3 − t)QB
+n−3
+′(t)
+�
+.
+Plugging it into eq. (11) and simplifying completes the proof.
+Based on the obtained recursion eq. (10), the first few polynomials of QB
+n (t) are com-
+puted and listed below:
+QB
+1 (t) =t + 1
+QB
+2 (t) =t2 + 4t + 1
+QB
+3 (t) =3t3 + 14t2 + 14t + 3
+QB
+4 (t) =11t4 + 64t3 + 112t2 + 64t + 11
+QB
+5 (t) =53t5 + 362t4 + 866t3 + 866t2 + 362t + 53
+QB
+6 (t) =309t6 + 2428t5 + 7252t4 + 10300t3 + 7252t2 + 2428t + 309
+QB
+7 (t) =2119t7 + 18806t6 + 66854t5 + 121838t4 + 121838t3 + 66854t2 + 18806t + 2119
+QB
+8 (t) =16687t8 + 165016t7 + 677656t6 + 1497880t5 + 1937368t4 + 1497880t3
++ 677656t2 + 165016t + 16687
+QB
+9 (t) =148329t9 + 1616786t8 + 7513658t7 + 19444106t6 + 30752450t5 + 30752450t4
++ 19444106t3 + 7513658t2 + 1616786t + 148329
+Corollary 8. Let F(x, t) = �
+n≥1
+QB
+n (t)xn be the generating function of QB
+n (t).
+Then,
+F(0, t) = 0 and F(x, t) satisfies the following differential equation:
+∂F
+∂t (x, t) + t + 1 + 3tx + x + 2tx2
+t(t2 − 1) + 2t2(t − 1)x
+∂F
+∂x (x, t)
+=
+−1 − t − 2tx
+t(t2 − 1)x + 2t2(t − 1)x2 −
+tx2 − 1
+t(t2 − 1)x2 + 2t2(t − 1)x3 F(x, t).
+(12)
+The proof of Corollary 8 is provided in the appendix. Unfortunately, we are unable to
+solve the differential equation to get explicit formulas for F(x, t) and QB
+n (t).
+Corollary 9. Let π ∈ QB
+n be chosen uniformly at random. Then, the expectation and
+variance of the number of signed elements b(π) are respectively
+E[b(π)] = n
+2,
+Var[b(π)] = Fn + 2n − n2
+4
+,
+where Fn satisfies
+Fn =
+�
+(n − 1)2 + (2n − 2)Fn−1
+�QB
+n−1
+QB
+n
++
+�
+(3n − 2)(n − 2) + (4n − 3)Fn−2
+�QB
+n−2
+QB
+n
++
+�
+(2n − 2)(n − 3) + (2n − 2)Fn−3
+�QB
+n−3
+QB
+n
+.
+7
+
+Proof. Recall that qn,m = qn,n−m, and it is easy to see
+QB
+n (1) =
+n
+�
+m=0
+qn,m,
+QB
+n
+′(t) =
+n
+�
+m=0
+mqn,mtm−1,
+QB
+n
+′(1) =
+n
+�
+m=0
+mqn,m,
+QB
+n
+′′(t) =
+n
+�
+m=0
+m(m − 1)qn,mtm−2,
+QB
+n
+′′(1) =
+n
+�
+m=0
+m(m − 1)qn,m.
+Consequently, we have
+E[b(π)] =
+�n
+m=0 mqn,m
+�n
+m=0 qn,m
+=
+�n
+m=0(m + n − m)qn,m/2
+�n
+m=0 qn,m
+= QB
+n
+′(1)
+QB
+n (1) = n
+2.
+As for the variance, we compute
+Var[b(π)] =
+n�
+m=0
+(m − E[b(π)])2qn,m
+QB
+n
+=
+n�
+m=0
+m2qn,m +
+n�
+m=0
+E[b(π)]2qn,m − 2
+n�
+m=0
+mE[b(π)]qn,m
+QB
+n
+=
+n�
+m=0
+�
+m(m − 1) + m
+�
+qn,m +
+n�
+m=0
+E[b(π)]2qn,m − 2
+n�
+m=0
+mE[b(π)]qn,m
+QB
+n
+= QB
+n
+′′(1) + QB
+n
+′(1) + E[b(π)]2QB
+n (1) − 2E[b(π)]QB
+n
+′(1)
+QB
+n (1)
+= QB
+n
+′′(1)
+QB
+n (1) + 2n − n2
+4
+.
+From Theorem 7, we next get
+QB
+n
+′′(1) = (2n − 2)QB
+n−1
+′(1) + (2n − 2)QB
+n−1
+′′(1) + (6n − 4)QB
+n−2
+′(1) + (4n − 3)QB
+n−2
+′′(1)
++ (4n − 4)QB
+n−3
+′(1) + (2n − 2)QB
+n−3
+′′(1).
+By dividing both sides by QB
+n , the following recurrsion of Fn = QB
+n
+′′(1)
+QB
+n (1) can be obtained:
+Fn =
+�
+(n − 1)2 + (2n − 2)Fn−1
+�QB
+n−1(1)
+QB
+n (1) +
+�
+(3n − 2)(n − 2) + (4n − 3)Fn−2
+�QB
+n−2(1)
+QB
+n (1)
++
+�
+(2n − 2)(n − 3) + (2n − 2)Fn−3
+�QB
+n−3(1)
+QB
+n (1) .
+This completes the proof.
+8
+
+The following corollary follows from Theorem 7 as well.
+Corollary 10. For n ≥ 3 and m ≥ 0, we have
+Qn =(n − 1)Qn−1 + (n − 2)Qn−2,
+(13)
+QB
+n =(2n − 1)QB
+n−1 + (2n − 4)QB
+n−2,
+(14)
+qn,m =(n − 1)qn−1,m−1 + (n − 1)qn−1,m + (m − 2)qn−2,m−2 + (3n − 5)qn−2,m−1
++ (n − m − 2)qn−2,m + (2m − 4)qn−3,m−2 + (2n − 2m − 4)qn−3,m−1,
+(15)
+where we make the convention that qn,m = 0 if m < 0.
+Proof. Eq. (13) and (14) follow from eq. (10) by setting t = 0 and t = 1, respectively.
+Eq. (15) is obtained by comparing the coefficients of tm on both sides of eq. (10)
+It is easy to see that the case m = 0 of eq. (15) agrees with eq. (13). Of course,
+eq. (13) and (14) can be also obtained by making use of the recursions satisfied by Dn,
+DB
+n , eq. (2) and eq. (3). We leave the computation to the interested reader. In the next
+section, we will present a direct combinatorial proof of the recursion of qn,m.
+3
+Recursion and unimodality of qn,m
+The goal of this section is to first prove the recursion of qn,m combinatorially, and then
+prove the sequence of qn,m is unimodal.
+Before we proceed, we present a connection to the work of the first author [3] using a
+slight variation of signed relative derangements. Recall the definitions there: Let
+Γn = {(0, −1), (−1, 0), (1, −2), (−2, 1), . . ., (n, −n − 1), (−n − 1, n)}
+be a set of ordered pairs. For an ordered pair T = (a, b), the element a is called the left
+entry of T and denoted by T l = a, while b the right entry of T and denoted by T r = b. A
+signed relative derangement (SRD) on Γn is a sequence π = T0T1 · · · Tn such that Ti ∈ Γn,
+each ordered pair appears at most once in π, (a, b) ∈ Γn and (b, a) ∈ Γn cannot be both
+contained in π, and for 0 ≤ i ≤ n − 1, T r
+i ̸= −T l
+i+1. This particular form for SRDs was
+chosen for a reason, as SRDs were also treated as fixed point involutions in [3]. As such,
+the first author could provide an upper bound for the number of signed permutations
+whose reversal distances are maximum possible.
+An SRD of type 1 on Γn is an SRD π = T0T1T2 · · · Tn such that T0 = (0, −1) and
+Tn ̸= (n, −n − 1). An SRD of type 2 on Γn is an SRD π = T0T1T2 · · ·Tn such that
+T0 = (0, −1) and Tn = (n, −n − 1). Let fn and ˆfn denote the number of SRDs of type 1
+and type 2 on Γn, respectively. Clearly, ˆfn = fn−1. One of the main results in Chen [3] is
+the four-term recursion below
+fn = (2n − 2)fn−1 + (4n − 3)fn−2 + (2n − 2)fn−3,
+(16)
+where f1 = 1, f2 = 4, f3 = 25.
+9
+
+Following [3], we have known that there is a natural bijection for transfroming SRDs
+on Γn to the signed relative derangements in the classical definition. That is, just view
+(i, −i−1) as i and (−i−1, i) as i. But it is worth noting that the condition now becomes
+that i is not followed by i + 1 and i + 1 is not followed by i.
+Sometimes it is more
+convenient to use this definition. For instance, let π[r] denote the sequence obtained from
+π by reading π reversely (i.e., right to left) and changing i to i and vice versa. Then, if π
+is an SRD, then π[r] is also an SRD. For example, for an SRD π = 2310, π[r] = 0132 is an
+SRD too. We refer to π[r] as the conjugate-reverse of π. This is not true in the classical
+definition. For example, for a signed relative derangement π = 3210, π[r] = 0123 is not a
+signed relative derangement anymore in the classical definition. In the following, we will
+use the new version of SRDs if not explicitly stated otherwise.
+Lemma 11. For n ≥ 3,
+QB
+n = (fn + fn−1) + (fn−1 + fn−2).
+(17)
+Proof. The elements π1π2 · · · πn in QB
+n consist of two classes: π1 = 1 and π1 ̸= 1. The
+latter is equivalent to SRDs of type 1 and type 2 and counted by fn + ˆfn = fn + fn−1 as
+discussed above. As for those starting with 1, the subsequence π2 · · ·πn must not start
+with 2. It is then not hard to see that this class is counted by fn−1 + fn−2, completing
+the proof.
+In view of Lemma 11, the “core” of QB
+n is really the subset of sequences not starting
+with 1. Also, recall that QB
+n = DB
+n + DB
+n−1 obtained by Chen and Zhang [5]. Accordingly,
+it suggests the following relation which can be viewed as a dual of this relation.
+Proposition 12 (Dual of eq. (3)). For n ≥ 2, we have
+DB
+n = fn + fn−1.
+(18)
+Proof. First, we take the opportunity to present a direct combinatorial proof of a recursion
+of DB
+n which is an analogue of eq. (1). Consider signed derangements of length n in DB
+n .
+We distinguish the following cases.
+case 1: If 1 appears, it can be placed at any other n − 1 positions except the first
+position. Suppose 1 is placed at the k-th position for a fixed 1 < k ≤ n, then we consider
+the elements k and k.
+• If k is placed at the first position, the remaining n − 2 entries (other than the first
+and the k-th entries) could essentially form any signed derangement of length n−2.
+Then, we have DB
+n−2 signed derangements in this case.
+• If k is not placed at the first position (note that k could still be placed at the first
+position), viewing k as 1 (and k as 1), the remaining n − 1 entries other than the
+k-th entry essentially form a signed derangement of length n − 1. Hence, there are
+DB
+n−1 signed derangements in this case.
+10
+
+Since there are n − 1 options for k, we have (n − 1)(DB
+n−2 + DB
+n−1) signed derangements
+where 1 appears.
+case 2: Consider the case 1 appears.
+• Clearly, there are DB
+n−1 signed derangements where 1 is placed at the first position.
+• If 1 is not placed at the first position, in analogy with case 1, we have (n−1)(DB
+n−2+
+DB
+n−1) such signed derangements.
+Summarizing the above discussion, we have
+DB
+n = (2n − 1)DB
+n−1 + (2n − 2)DB
+n−2.
+(19)
+Next, let Fn = fn + fn−1. Applying the four-term recurrence eq. (16), we have
+Fn = (2n − 1)fn−1 + (4n − 3)fn−2 + (2n − 2)fn−3
+= (2n − 1)Fn−1 + (2n − 2)Fn−2.
+That is, DB
+n and Fn satisfy the same recursion.
+Meanwhile, we have DB
+2 = F2 = 5,
+DB
+3 = F3 = 29. Therefore, DB
+n and fn + fn−1 also have the same initial values. Thus, it
+is proved that DB
+n = fn + fn−1.
+We remark that eq. (19) can be found in [2], but with a different proof. Combining
+Proposition 12 and eq. (16), we immediately have an alternative proof of eq. (14).
+Now we are in a position to prove the recursion eq. (15). Let qn,m denote the number
+of π = π1π2 · · · πn ∈ QB
+n with m bar-elements and π1 ̸= 1. Equivalently, qn,m counts SRDs
+of type 1 and 2 on Γn that have m bar-elements. We first have the following relation
+which is an analogue of eq. (17).
+Lemma 13.
+qn,m = qn,m + qn−1,m.
+(20)
+Proof. For any π ∈ QB
+n with m bar-elements, π is either in the form π1π2 · · ·πn where
+π1 ̸= 1 or 1π2 · · ·πn. The number of the former is just qn,m. And the number of the latter
+is equal to the number of π2 · · · πn where π2 ̸= 2, namely qn−1,m, whence the lemma.
+In the light of Lemma 13, in order for studying qn,m it suffices to study qn,m. To that
+end, we generalize the idea for proving eq. (16) in [3] and obtain
+Theorem 14. For n ≥ 4, we have
+qn,m =(n − 1)qn−1,m + (n − m − 1)qn−2,m + (m − 1)qn−2,m−1 + nqn−1,n−m
++ (m − 1)qn−2,n−m + (n − m − 1)qn−2,n−m−1.
+(21)
+11
+
+Proof. Note that SRDs of type 1 and 2 on Γn with m bar-elements are either in the form
+0A11A2 or 0A11A2. We will count SRDs in each case separately.
+case 1: 0A11A2.
+(i) A2 = ∅. In this case, A1 could essentially be any SRD of length n − 1 with m
+bar-elements. It is easy to see there are qn−1,m + qn−2,m such SRDs.
+(ii) A2 ̸= ∅. Consider the induced sequence 1A2A1.
+If there exists no a ∈ [n] such that A2 ends with a while A1 starts with a − 1 or A2
+ends with a − 1 while A1 starts with a, then the sequence 1A2A1 could be equivalently
+any SRD of type 1 or 2 of length n−1 and with m bar-elements. The latter is counted by
+qn−1,m. Moreover, there are n − 2 ways to transform each such a sequence into sequences
+of the form A11A2. Hence, there are (n − 2)qn−1,m SRDs lying in this situation.
+If otherwise, such an a exists, then by construction a ∈ [n] \ [2]. We claim that for a
+fixed a ∈ [n] \ [2],
+• the sequences of the form 1A′
+2aa − 1A′
+1 are in one-to-one correspondence to the
+SRDs on the set Γn−1 \ {0, 0} starting with 1 and having m − 1 bar-elements which
+are counted by qn−2,m−1;
+• the sequences of the form 1A′
+2(a − 1)aA��
+1 are in one-to-one correspondence to the
+SRDs on the set Γn−1 \ {0, 0} starting with 1 and having m bar-elements which are
+counted by qn−2,m.
+The above first case can be seen from replacing aa − 1 with a − 1 and decreasing all other
+elements greater than a (regardless of if it has a bar) by 1. In particular, this will lose one
+bar-element. The second case can be seen analogously, but without losing a bar-element.
+Conversely, for each of the m − 1 bar-elements in the SRDs on the set Γn−1 \ {0, 0}
+starting with 1, say a − 1 (a > 2), we first increase all elements no less than a by one, and
+then replace a − 1 with aa − 1. Clearly, the resulting sequence is of the form 1A′
+2aa − 1A′
+1.
+In addition, there is a unique way to transform such a sequence into an SRD of the form
+0A11A2, i.e., 0a − 1A′
+11A′
+2a. So, there are (m − 1)qn−2,m−1 SRDs lying in this situation.
+Analogously, we find there are (n − 2 − m)qn−2,m SRDs of the form 0aA′
+11A′
+2(a − 1).
+In summary, for n ≥ 4, the number of SRDs of type 1 and type 2 with m bar-elements
+on Γn in the form 0A11A2 is given by
+(n − 1)qn−1,m + (n − m − 1)qn−2,m + (m − 1)qn−2,m−1.
+case 2: 0A11A2. Consider the induced sequence 1A[r]
+1 A[r]
+2 first (Recall A[r]
+i
+denotes the
+conjugate-reverse of Ai). Apparently, there are n − m bar-elements in A[r]
+1 A[r]
+2 .
+(i) A[r]
+1 = ∅.
+In this scenario, A[r]
+2
+could essentially be any SRD of length n − 1 with n − m bar-
+elements the number of which is given by qn−1,n−m + ¯qn−2,n−m.
+(ii) A[r]
+1 ̸= ∅.
+12
+
+When A[r]
+2 = ∅, 1A[r]
+1 is the conjugate-reverse of A11 thus is an SRD of length n − 1.
+Consequently, the number of SRDs in this case is qn−1,n−m.
+Suppose A[r]
+2 ̸= ∅. Similar to case 1 (ii), there are (n − 2)qn−1,n−m SRDs where there
+is no a ∈ [n] such that A[r]
+1
+ends with a while A[r]
+2
+starts with a − 1 or A[r]
+1
+ends with
+a − 1 while A[r]
+2 starts with a. Suppose otherwise such an a exists. For a fixed a ∈ [n]/[2],
+similar to the discussion in case 1 (ii), we claim that
+• the sequences of the form 1A[r]
+1
+′aa − 1A[r]
+2
+′ are in one-to-one correspondence to the
+SRDs on the set Γn−1 \ {0, 0} starting with 1 and having n − m − 1 bar-elements
+which are counted by (n − m − 1)qn−2,n−m−1;
+• the sequences of the form 1A[r]
+1
+′(a−1)aA[r]
+2
+′ are in one-to-one correspondence to the
+SRDs on the set Γn−1 \ {0, 0} starting with 1 and having n − m bar-elements which
+are counted by (m − 2)qn−2,n−m.
+In summary, for n ≥ 4, the number of SRDs of type 1 and type 2 with m bar-elements
+on Γn in the form 0A11A2 is given by
+nqn−1,n−m + (m − 1)qn−2,n−m + (n − m − 1)qn−2,n−m−1.
+Combining the above two cases together, the theorem follows.
+Applying Theorem 14, we have
+qn,m =qn,m + qn−1,m
+=(n − m − 1)qn−1,m + (n − m − 2)qn−2,m + mqn−1,m + (m − 1)qn−2,m−1
++ (m − 1)qn−1,m−1 + (m − 1)qn−2,m−1 + (n − m + 1)qn−1,n−m
++ (n − m − 1)qn−2,n−m−1 + (n − m)qn−2,n−m−1 + (n − m − 2)qn−3,n−m−2,
+and
+qn−1,m−1 =qn−1,m−1 + qn−2,m−1
+=(n − m − 1)qn−2,m−1 + (n − m − 2)qn−3,m−1 + (m − 1)qn−2,m−1
++ (m−2)qn−3,m−2 + (m−2)qn−2,m−2 + (m−2)qn−3,m−2 + (n−m+1)qn−2,n−m
++ (n − m − 1)qn−3,n−m−1 + (n − m)qn−3,n−m−1 + (n − m − 2)qn−4,n−m−2.
+Summing up the above two equations, we can clear all numbers of the form qx,y and arrive
+at
+qn,m + qn−1,m−1 =nqn−1,m−1 + (n − 1)qn−1,m + (m − 2)qn−2,m−2 + (3n − 5)qn−2,m−1
++ (n − m − 2)qn−2,m + (2m − 4)qn−3,m−2 + (2n − 2m − 4)qn−3,m−1.
+Moving qn−1,m−1 to the right-hand side, we obtain eq. (15) as desired.
+13
+
+Is it true that there will be more signed relative derangements if we turn more unsigned
+elements into signed elements? Put it differently, is it easier to form a relative derangement
+if more elements have signs? The answer is apparently negative due to the symmetry of
+qn,m. But, how about the cases for m ≤ n/2? This is related to the unimodality of
+sequences. The sequence x0, x1, x2, · · · , xn is said to be unimodal if there exists an index
+0 ≤ m ≤ n, called the mode of the sequence, such that x0 ≤ · · · ≤ xm−1 ≤ xm ≥ xm+1 ≥
+· · · ≥ xn.
+Theorem 15. For any fixed n ≥ 1, the sequence qn,0, qn,1, . . . , qn,n is unimodal.
+Proof. Thanks to the symmetry of qn,m, it suffices to prove P(n, m) = qn,m − qn,m−1 ≥ 0
+for m ≤ n/2, where we still make the convention qn,m = 0 if m < 0. We shall prove this
+mainly by induction.
+First, from the polynomials of QB
+n (t) listed in the last section, we observe that for
+n = 1, 2, . . . , 9 and m ≤ n/2, P(n, m) ≥ 0. Secondly, we claim
+• for any n ≥ 2, P(n, 1) ≥ 0;
+• for any n ≥ 4, P(n, 2) ≥ 0.
+In order for proving P(n, 1) ≥ 0 in the case of n ≥ 2, we construct an injection from
+Qn to QB
+n,1 (where QB
+n,i denotes the subset containing signed relative derangements with i
+bar-elements). For each sequence in Qn, replacing n with n, we obtain a unique sequence
+in QB
+n,1. Obviously, this is an injection and then P(n, 1) ≥ 0 follows.
+Analogously, we construct an injection from QB
+n,1 to QB
+n,2 for proving P(n, 2) ≥ 0. We
+will classify the sequences in QB
+n,1 by the largest bar-element.
+case 1: If the largest bar-element in QB
+n,1 is less than n − 1, then we map it to a
+relative derangement obtained by substituting n for n. In this case, the obtained relative
+derangements in QB
+n,2 have two bar-elements: n and i for some 1 ≤ i < n − 1.
+case 2: If the largest bar-element in QB
+n,1 is exactly n − 1, and n − 1 is not followed
+by n, then we substitute n for n. In the case that n − 1 is followed by n, we replace 1
+with 1 to obtain a sequence in QB
+n,2. In this case, the obtained relative derangements in
+QB
+n,2 have two bar-elements: either n − 1 and n, or n − 1 and 1 with an additional feature
+that n − 1 is followed by n.
+case 3: Suppose the largest bar-element in QB
+n,1 is n. If n−1 is not followed by n, then
+we remove the bar of n. Meanwhile, we replace n−1 with n − 1 and 1 with 1. If n follows
+n − 1, then we simply replace 1 with 1. In this case, the obtained relative derangements
+in QB
+n,2 have two bar-elements: either n − 1 and 1 with an additional feature that n − 1
+is not followed by n, or n and 1 with the feature that n follows n − 1.
+In the above mapping procedure, relative derangements in QB
+n,1 lying in the same
+case are clearly mapped to distinct relative derangements in QB
+n,2. Moreover, inspecting
+the patterns of the contained two bar-elements and the additional features, relative de-
+rangements from different cases are mapped to distinct relative derangements in QB
+n,2 (for
+n ≥ 4) as well. Therefore, the above map is indeed an injection. Hence, P(n, 2) ≥ 0.
+14
+
+Now suppose for 1 ≤ n ≤ N and any 0 ≤ m ≤ n/2, P(n, m) ≥ 0. Next, we shall show
+that P(N + 1, m) ≥ 0 for any 3 ≤ m ≤ (N + 1)/2. Applying Corollary 10, we first have
+P(N + 1, m) = qN+1,m − qN+1,m−1
+=N(qN,m − qN,m−2) + (N − m − 1)(qN−1,m − qN−1,m−1)
++ 3(N − 1)(qN−1,m−1 − qN−1,m−2) + (m − 3)(qN−1,m−2 − qN−1,m−3)
++ 2(N − m − 1)(qN−2,m−1 − qN−2,m−2) + 2(m − 3)(qN−2,m−2 − qN−2,m−3).
+(22)
+We proceed to distinguish two cases.
+(i) If 3 ≤ m ≤ (N − 1)/2, we compare the two subscripts of each term qx,y on the RHS
+of eq. (22) and find that y ≤ x/2. For instance, since the maximum value of m here is
+(N − 1)/2, as to qN−1,m−2, we have m − 2 = (N − 5)/2 which satisfies (N − 1)/2 ≥ m − 2.
+Consequently, qN−1,m−2 −qN−1,m−3 ≥ 0 by assumption. Other summands are nonnegative
+by the same token. Therefore, P(N + 1, m) ≥ 0 follows.
+(ii) If N/2 ≤ m ≤ (N + 1)/2, m equals either N/2 or (N + 1)/2 since m ∈ N. We check
+the two subscripts of qx,y and find that y > x/2 in some cases. Therefore, in the following
+reasoning, we will make some transformation by the symmetry of qn,m.
+When m = N/2, we replace qN−1,m with qN−1,N−m−1 and regroup the terms on the
+RHS of eq. (22), and obtain
+P(N + 1, m) =N(qN, N
+2 − qN, N−4
+2 ) + 3(N − 1)(qN−1, N−2
+2
+− qN−1, N−4
+2 )
++ N − 6
+2
+(qN−1, N−4
+2
+− qN−1, N−6
+2 ) + (N − 2)(qN−2, N−2
+2
+− qN−2, N−4
+2 )
++ (N − 6)(qN−2, N−4
+2
+− qN−2, N−6
+2 ).
+(23)
+Similarly, when m = (N + 1)/2, we replace qN,m with qN,N−m, qN−1,m with qN−1,N−m−1
+and qN−2,m−1 with qN−2,N−m−1 in eq. (22) and regroup the terms to have
+P(N + 1, m) =N(qN, N−1
+2
+− qN, N−3
+2 ) + 5N − 3
+2
+(qN−1, N−1
+2
+− qN−1, N−3
+2 )
++ N − 5
+2
+(qN−1, N−3
+2
+− qN−1, N−5
+2 ) + (N − 5)(qN−2, N−3
+2
+− qN−2, N−5
+2 ).
+(24)
+Inspecting term by term on the RHS of eq. (23) and eq. (24), they are all nonnegative by
+assumption. Therefore, P(N + 1, m) ≥ 0. This completes the proof of the theorem.
+Acknowledgements
+The authors would like to thank Prof. Yi Wang for pointing out that the roots of QB
+n (t)’s
+are not necessarily all real.
+A
+Proof of Corollary 2.13
+In the following, we write ∂F
+∂x (x, t) as Fx(x, t) and ∂F
+∂t (x, t) as Ft(x, t). Then according to
+15
+
+the definition of F(x, t), we first have
+Fx(x, t) =
+�
+n≥1
+nQB
+n (t)xn−1, Ft(x, t) =
+�
+n≥1
+QB
+n
+′(t)xn.
+For the terms on right-hand side of eq. (10), multiplying by xn and summing over n ≥ 3,
+we respectively obtain
+�
+n≥3
+(n − 1)tQB
+n−1(t)xn = tx2 �
+n≥3
+(n − 1)QB
+n−1(t)xn−2
+= tx2(Fx(x, t) − QB
+1 (t))
+�
+n≥3
+(n − 1)QB
+n−1(t)xn = x2 �
+n≥3
+(n − 1)QB
+n−1(t)xn−2
+= x2(Fx(x, t) − QB
+1 (t))
+�
+n≥3
+(t3 − t)QB
+n−2
+′(t)xn = x2(t3 − t)
+�
+n≥3
+QB
+n−2
+′(t)xn−2
+= x2(t3 − t)Ft(x, t)
+�
+n≥3
+(3n − 5)tQB
+n−2(t)xn = t
+� �
+n≥3
+3nQB
+n−2(t)xn − 5
+�
+n≥3
+QB
+n−2(t)xn�
+= t
+� �
+n≥3
+3(n − 2 + 2)QB
+n−2(t)xn − 5
+�
+n≥3
+QB
+n−2(t)xn�
+= t
+�
+3
+�
+n≥3
+(n − 2)QB
+n−2(t)xn + 6
+�
+n≥3
+QB
+n−2(t)xn − 5
+�
+n≥3
+QB
+n−2xn�
+= t
+�
+3x3 �
+n≥3
+(n − 2)QB
+n−2(t)xn−3 + x2 �
+n≥3
+QB
+n−2(t)xn−2�
+= t
+�
+3x3Fx(x, t) + x2F(x, t)
+�
+�
+n≥3
+(n − 2)QB
+n−2(t)xn = x3 �
+n≥3
+(n − 2)QB
+n−2(t)xn−3 = x3Fx(x, t)
+�
+n≥3
+(2t3 − 2t2)QB
+n−3
+′(t)xn = (2t3 − 2t2)x3 �
+n≥3
+QB
+n−3
+′(t)xn−3
+= (2t3 − 2t2)x3Ft(x, t)
+16
+
+�
+n≥3
+(2n − 6)tQB
+n−3(t)xn = t
+� �
+n≥3
+2nQB
+n−3(t)xn − 6
+�
+n≥3
+QB
+n−3(t)xn�
+= t
+�
+2
+�
+n≥3
+(n − 3 + 3)QB
+n−3(t)xn − 6
+�
+n≥3
+QB
+n−3(t)xn�
+= t
+�
+2
+�
+n≥3
+(n − 3)QB
+n−3(t)xn�
+= t
+�
+2x4 �
+n≥3
+(n − 3)QB
+n−3(t)xn−4�
+= 2tx4Fx(x, t)
+According to the computation above, for n ≥ 3, we have
+�
+n≥3
+QB
+n (t)xn =tx2(Fx(x, t) − QB
+1 (t)) + x2(Fx(x, t) − QB
+1 (t)) + x2(t3 − t)Ft(x, t)
++ t
+�
+3x3Fx(x, t) + x2F(x, t)
+�
++ x3Fx(x, t)
++ (2t3 − 2t2)x3Ft(x, t) + 2tx4Fx(x, t)
+=
+�
+(t + 1)x2 + (3t + 1)x3 + 2tx4�
+Fx(x, t)
++
+�
+(t3 − t)x2 + (2t3 − 2t2)x3�
+Ft(x, t) + tx2F(x, t) − (t + 1)2x2.
+Then, F(x, t) is given as follows:
+F(x, t) =QB
+1 (t)x + QB
+2 (t)x2 +
+�
+n≥3
+QB
+n (t)xn
+=x + tx + t2x2 + 4tx2 + x2 +
+�
+(t + 1)x2 + (3t + 1)x3 + 2tx4�
+Fx(x, t)
++
+�
+(t3 − t)x2 + (2t3 − 2t2)x3�
+Ft(x, t) + tx2F(x, t) − (t + 1)2x2
+=
+�
+(t + 1)x2 + (3t + 1)x3 + 2tx4�
+Fx(x, t)
++
+�
+(t3 − t)x2 + (2t3 − 2t2)x3�
+Ft(x, t) + tx2F(x, t) + (t + 1)x + 2tx2.
+After sorting out the above equations, we eventually obtain
+Ft(x, t) + t + 1 + (3t + 1)x + 2tx2
+t(t2 − 1) + 2t2(t − 1)x Fx(x, t) +
+tx2 − 1
+t(t2 − 1)x2 + 2t2(t − 1)x3F(x, t)
+=
+−1 − t − 2tx
+t(t2 − 1)x + 2t2(t − 1)x2,
+completing the proof of Corollary 8.
+References
+[1] C.-O. Chow. On derangement polynomials of type B. S´em. Lothar. Combin., 55:Art.
+B55b, 2006.
+17
+
+[2] C.-O. Chow. On derangement polynomials of type B. II. J. Combin. Theory Ser. A,
+116(4):816–830, 2005.
+[3] R.X.F. Chen. Signed relative derangements, reversal distance, and the signed Hult-
+man numbers. Ars. Combin., 147:281–288, 2019.
+[4] W.Y.C. Chen. The skew, relative, and classical derangements. Discrete Math.,
+160:235–239, 1996.
+[5] W.Y.C. Chen and J.C.Y. Zhang. The skew and relative derangements of type B.
+Electron. J. Combin., 14:2147–2153, 2007.
+[6] W.Y.C. Chen, R. L. Tang and A.F.Y. Zhao. Derangement polynomials and ex-
+cedances of type B. Electron. J. Combin., 16(2):R15, 2009.
+[7] L.L. Liu and M. Dong. Cyclic derangement polynomials of the wreath product Cr≀Sn.
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+[8] R.P. Stanley. Enumerative Combinatorics, vol. 1. Cambridge University Press, Cam-
+bridge, 1997.
+[9] M.L. Wachs. On q-derangement numbers. Proc. Amer. Math. Soc., 106(1):273–278,
+1989.
+[10] A.F.Y. Zhao. Excedance numbers for the permutations of type B. Electron. J. Com-
+bin., 20(2):P28, 2013.
+18
+

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